Digitized  by  the  Internet  Archive 

in  2008  with  funding  from 

Microsoft  Corporation 


http://www.archive.org/details/commonschoolaritOOeatorich 


THE 


COMMON  SCHOOL 

ARITHMETIC 


1NINO 


ANALYSIS  AND  SYNTHESIS ; 

APAPTEU  TO 

THE  BEST  MODE  OF  INSTRUCTION  IN  THE  ELEMENTS 
OF  WRITTEN  ARITHMETIC. 


BY 

JAMES  S.  EATOX,  M.  A., 

IKiTBrCTOR    I!»    rHILLlPS    ACADEMY,     AXDOVER,     AND    AUTHOR    OF    "EA*Y    I.MIOJI    IIT 
Mi.STAL    ARITHMETIC,"    A>D    "A    TREATISE    OX    WRITTEN    AH*  ruMETlC.- 


[fc     BOSTOWA     $) 
S&.      1630.      jSt 


BOSTON: 
1  AGGARD      AND      THOMPSON, 

20  COKNHILL. 
SAN  FRANCISCO:  II.  II.  BANCROFT  *  COMPANY. 

18 


•     •  .    • 


Entered,  according  to  Act  of  Congress,  in  the  year  1863, 

By  JAMES  S.  EATON,  M.  A., 

In  the  Clerk's  Oflico  of  the  District  Court  of  the  District  of  Massachusetts 


EDUCAT1UW  DKPT 


A  .V  D  O  V  r.  R  : 
ILBCIEOirriD      A5D      P  K  I  Jf  T  X  D 
JB  T     W.     r.     U  E  A  P  K  K. 


PB K FACE 


Thf.rk  is  a   largo  class  of   pupils  whose   limited  time  renders  it 

lible    for   them   to  pursue   an    extended   mathematical  course. 

The  author,  in  accordance  with  his  original   intention  to  prepare  a 

oki    in    Arithmetic,  has   now  endeavored    to    adapt 

this  work  to  tho  wants  of  this  class  of  pupils. 

With  this  purpose  in  view,  the  simple,  elementary,  practical  prin- 
ciples of  the  science   are   more  fully  presented   than  in   his   larger 
work,  while  the  more  intricate  and  less  important  parts  have  been 
I   more  briefly  or  entirely  omitted.     A  corresponding   change 
in  the  character  of  the  examples  has  also  been  made. 

As  in  the  irork,  so  here,  constant  attention  has  been  paid 

to  the  brevity,  simplicity,  perspicuity,  and  accuracy  of  expression  ; 
and    no    effort    has    been    spared    in    the  endeavor   to    render    the 
mica!  execution  appropriate  and  attractive, 
nitions,  tables,  and    explanations   of   signs  have  been  distrib- 
through    the  book  where   their   aid    is  needed,  to    enable    the 
pupil    to    learn    them    more    readily  than   when  they  are    pre 
ivfly. 

ily  all  the  example!  have  been  prepared  for  this  book,  and 
are  different  from  those  of  the  larger  work  ;  still,  to  secure  uniform- 
ity of  language  (a  matter  of  great  importance,  as  every  experi 

-.  the    leading    example!  in   the    several  subjects,  the 
definitions  and    rules,  with  few   exceptions,  have  been  intentionally 
1   with  but  little  modification. 

6*349529 


iv  PKEFACE. 

Articles  on  United  States  Money,  Percentage,  Stocks,  Custom- 
House  Business,  and  Exchange  have  been  prepared  for  this  book ; 
and  all  the  principles  requisite  for  a  practical  business  life  have 
been  presented  in  a  simple,  intelligible,  attractive  manner,  and 
with  sufficient  minuteness  and  fullness  and  a  due  regard  tc  logical 
arrangement 

Brief,  suggestive  questions  have  been  placed  at  the  bottom  o( 
the  page,  designed  in  no  way  to  interfere  with  the  fvwt  original 
questioning  which  every  teacher  will  adopt  for  himself,  but  merely 
to  aid  the  young  and  inexperienced  pupil  in  fixing  his  attention 
upon  the  more  important  parts  of  the  subject 

Here,  as  in  the  larger  work,  some  of  the  answers  to  examples 
have  been  given  to  inspire  confidence  in  the  learner,  and  others 
are  omitted  to  secure  the  discipline  resulting  from  proving  the  oper- 
ations, a  discipline  and  a  benefit  which  the  pupil  should  not  forego 
nor  the  teacher  neglect 

Fully  appreciating  the  favor  which  has  been  bestowed  on  his 
other  works,  the  author  sends  this  forth,  hoping  it  may  commend 
itself  to  the  approval  of  committees  and  teachers,  and  that  it  may 
be  found  adapted  to  contribute  in  6ome  measure  to  the  happiness 
and  improvement  of  the  class  of  pupils  for  whom  it  is  deagned. 

A  Key,  containing  the  Answers  not  given  in  this  book,  is  published 
for  the  use  of  Teachers. 

Phillips  Academy,  Andovek,  ) 
April  ld'}  18G2.  \ 


CONTENTS 


SIMPLE  NUMRERS. 


Definitions       

Notation  and  Numeration       .    . 

French  Numeration  Table       .    . 

-  in  French  Numeration 

.  Numeration  Table      .    . 

Exercises  in  English  Numeration 


PAOE 

.      7 


PAOK 

Roman  Notation 15 

K\«  irises  in  Roman  Notation    .    .  10 

Addition 17 

Subtraction 24 

Multiplication SO 

Division       42 


REDUCTION  OF  COMPOUND  NUMBERS. 


Definitions  58 

English  Money 69 

Weight 62 

Apothecaries'  Weight 63 

Avoirdupois  Weight 64 

Cloth  Measure 66 

Long  Measure 67 

Chain  Measure 68 


Square  Measure 69 

Solid  Measure 71 

Liquid  Measure 74 

Dry  Measure 75 

Time 70 

Circular  Measure 77 

Miscellaneous  Table 78 

Examples  in  Reduction      ....  79 


GENERAL  PRINCIPLES. 


Definitions       80 

Factoring  Numbers 81 


Greatest  Common  Divisor 
Least  Common  Multiple 


COMMON  FRACTIONS. 


General  Principles  of  Fractions      .    92 
Mixed   and    Whole   Numbers    Re- 
duced to  Improper  Fractions     .    93 
Improper    Fractions    Reduced    to 

Whole  or  Mixed  Numbers  .  .  95 
Fraction  Reduced  to  Lower  Terms  95 
Fraction  Multiplied  by  an  Integer  96 
Fraction  Divided  by  an  Integer  .  98 
Fraction  Multiplied  by  a  Fraction   100 

CMMcttng 101 

Fraction  Divided  by  a  Fraction     .  104 
Complex  Fractions  made  Simple    .  107 

Common  Denominator 108 

Common  Numerator 110 

1* 


To  Reduce  a  Fraction  of  a  Higher 

Denomination  to  one  of  a  Lower  111 
To  Reduce  a  Fraction  of  a  Lower 

Denomination  to  one  of  a  Higher  112 
To  Reduce  a  Fraction  of  a  Higher 
Denomination  to  Whole  Num- 
bers of  Lower  Denominations  .  113 
To  Reduce  Whole  Numbers  of 
Lower  Denominations  to  a  Frac- 
tion of  a  Higher  Denomination    114 

Addition  of  Fractions 116 

Subtraction  of  Fractions  .  .  .  .119 
Miscellaneous  Examples  ....  121 
Analysis 122 


VI 


CONTENTS. 


DECIMAL  FRACTIONS. 


PACE 

Definitions 128 

Decimal  Numeration  Table    ...  129 

Notation  and  Numeration      ...  131 

Addition 132 

Subtraction 133 

Multiplication 134 

Division 136 

Circulating  Decimals 137 


PAG) 

Common  Fractions  reduced  to  Dec- 
imals      138 

Integers  of  Lower  Denominations 
Seduced  to  the  Decimal  of  a 
Higher  Denomination    ....    140 

A  Decimal  of  a  Higher  Denomina- 
tion Reduced  to  Integers  of  Low- 
er Denominations 141 


UNITED    STATES    MONET. 


Definitions  and  Table 144 

Reduction 140 

Addition 147 

Subtraction 148 

Multiplication 148 


Division 148 

Practical  Examples 149 

Aliquot  Parts  of  a  Dollar  ....  152 

Bills 154 

Miscellaneous  Examples     ....  156 


COMPOUND    NUMBERS. 


Addition 158 

Subtraction 162 

Multiplication   • 166 


Longitude  and  Time 171 

Division 172 

Duodecimals 176 


PERCENTAGE. 


Definitons  and  Problems    .    .    .    .  1S3 

Interest 1«7 

Partial  Payments 194 

Problems  in  Interest 203 

Compound  Interest 206 

Discount 209 

Banking  and  Bank  Discount      .    .  211 

Insurance 214 


Stocks 216 

Commission  and  Brokerage   ...  219 

Taxes Ll'l 

Custom-House  Business      ....  224 

Exchange 227 

Equation  of  Payments 232 

Profit  and  Loss 242 

Partnership 248 


MISCELLANEOUS. 

Ratio 254  ;  Application  of  Square  Root 

Proportion 256 

Simple  Proportion     ......  257 

Compound  Proportion 263 

Alligation  Medial 268 

Alligation  Alternate 268 

Involution 274 

Evolution 276 

Square  Root 277 


Cube  Root 

Application  of  Cube  Root 
Arithmetical  Progression 
Geometrical  Progression 

Annuities 

Permutations  .... 
Mensuration  .... 
Miscellaneous  Examples 


282 
286 
291 
292 
295 
298 
300 
301 
307 


ARITHMETIC. 


ttcle  fl.    Arithmetic  is  the  science  of  numbers,  and  the 
art  of  computation* 

A  NiMi.ri;  ifl  :i  unit  or  a  collection  of  units,  a  unit  being 
one,  i.  e.  a  single  thing  of  any  kind ;  thus,  in  the  number  six 
tin-  unit  is  one  ;  in  ten  days  the  unit  is  one  day, 

2.    All  numbers  are  concrete  or  abstract. 

A  Concrete  Number  is  a  number  that  is  applied  to  a  par- 
ticular object ;  as  six  books,  ten  men,  four  days. 

An  Abstract  Number  is  a  number  that  is  not  applied  to 
any  particular  X)bject ;  as  six,  ten,  seventeen. 

3*    Arithmetic  employs  six  different  operations,  viz.  Notation, 
■u,  Addition,  Subtraction,  Multiplication,  and  Division. 


NOTATION   AND   NUMERATION. 

4r»    Notation  is  the  art  of  expressing  numbers  and  their 
relations  to  each  other  by  means  of  figures  and  other  symbols. 

•5.    NUMERATION  is  the  art  of  reading  numbers  which  have 
1   by  figun 

Art.  I.  What   is   Arithrmtir '     What  is  a  Number?    A  Unit?    %■  Wh»t  Is  s 
inber?    An  Abstract  Number?    3.    How  many  operations  in  Arith- 
metic?   What  »r«  the> •»    4.    What  is  Notation?    *.  Numeration? 


NOTATION   AND   NUMERATION. 

6.  Two  methods  of  notation  are  in  common  use  :  the  Arabic 
and  the  Roman. 

7.  The  Arabic  Notation,  or  that  brought  into  Europe  by 
the  Arabs,  employs  ten  figures  to  express  numbers,  viz. : 

0,        1,       2,        3,        4,       5,      6,       7,        8,        9. 
Naught,  One,   Two,   Three,  Four,  Five,   Six,   Seven,  Eight,  Nine. 

These  figures  are  called  digits,  from  the  Latin  digitus,  a 
finger ;  a  term  probably  applied  to  figures  from  the  custom  of 
counting  upon  the  fingers. 

8.  The  first  Arabic  figure,  0,  is  called  a  cipher,  naught,  or 
zero,  and,  standing  alone,  it  signifies  nothing. 

Each  of  the  remaining  nine  figures  represents  the  number 
placed  under  it,  and  for  convenience  in  distinguishing  them  from 
0,  they  are  called  significant  figures. 

9.  No  number  greater  than  nine  can  be  expressed  by  a 
single  Arabic  figure,  but  by  repeating  the  figures,  and  arranging 
them  differently,  all  numbers  may  be  represented. 

Ten  is  expressed  by  writing  the  figure  1  at  the  left  of  the 
cipher;  thus,  10.  In  like  manner,  twenty,  thirty,  forty,  etc., 
are  expressed  by  placing  2,  3,  4,  etc.,  at  the  left  of  0 ;  thus, 

20,         30,       40,      50,       60,         70,  80,         90. 

Twenty,    Thirty,    Forty,    Fifty,    Sixty,    Seventy,    Eighty,    Ninety. 

10.  The  numbers  from  10  to  20  are  expressed  by  placing 
the  figure  1  at  the  left  of  each  of  the  significant  figures ;  thus, 

11.  12,  13,  14,  15,         16,  17,        etc. 
Eleven,  Twelve,  Thirteen,  Four'.een,  Fifteen,  Sixteen,  Seventeen,  etc. 

In  a  similar  manner  .all  the  numbers,  up  to  one  hundred,  may 
be  Written;  thus, 

21,  36,  66,  98,  etc. 

Twenty-one,      Thirty-six,      Sixty-six,      Ninety-eight,      etc. 

6.  How  many  methods  of  Notation?  What?  7.  How  many  figures  in  the 
Arabic  Notation?  What  caJled?  Why?  8.  What  is  the  first  figure,  0,  called? 
The  others?  Why?  9.  The  largest  number  expressed  by  one  figure?  Ten, 
how  expressed?    Twenty?    10.  Numbers  from  ten  to  twenty,  how  expressed? 


NOTATION    AND    NIMEKATK  ».\.  S 

11.   One  hundred  \s  i  by  placing  the  figure  1  at  the 

of  too  ciphers;  thus    100,     In  like  manner  two  hundred, 

three  hundred,  etc.,  are  written;  thus, 

200,  300,  GOO,  800, 

Two  hundred,  Three  hundred,  Six  hundred,  Ei<;lit  hundred,  etc. 

1£.    The  other  numbers,  up  to  one  thousand,  may  be  ex- 
ed  by  putting  a  significant  figure  in  the  place  of  one  or 

cadi  of  the  ciphers  in  the  above  numbers  ;   thu*, 

Two  hundred  and  three,  expressed  in  figures,  is  203, 

hundred  and  eighty,  expressed  in  figures,  is  680, 

>.'ine  hundred  and  ninety-eight,  expressed  in  figures,  is  998. 

13.    The  place  of  a  figure  is  the  position  it  occupies  with 
iice  to  other  figures;  thus,  in  436,  the  G,  counting  from  the 
right,  is  in  the  Jirst  place,  3  is  in  the  second  place,  and  4  in  the 
third  place. 

The  figure  in  the  Jirst  place  represents  simple  units,  or  units 
of  the  first  order  ;  the  second  figure  represents  tens,  or  units  of 
the  second  order;  the  third,  hundreds,  or  units  of  the  third 
order ;  the  fourth,  thousands,  or  units  of  the  fourth  order,  etc. ; 
thus,  in  the  number  3576,  the  6  is  6  units  of  the  first  order; 
the  7  tens  is  7  units  of  the  second  order ;  the  5  hundreds  is  5 
'units  of  the  third  order,  etc. 

11.  From  the  foregoing  it  will  be  seen  that  each  significant 
figure  has  two  values ;  one  of  which  is  constant  (i.  e.  always  the 
same),  the  other  variable;  thus,  in  each  of  the  numbers  2,  20, 
and  200,  the  left  hand  figure  is  two  ;  but  in  the  first  it  is  two 
t/)iits  ;  in  the  second,  two  tens;  and  in  the  third,  two  hundreds. 

The  former  of  these  values  is  the  inherent  or  simple  value, 
and  the  latter  is  the  local  or  place  value. 

It5.  It  is  also  evident  that  the  value  of  a  figure  is  made  ten 
fold  by  removing  it  one  place  toward  the  left;  a  hundred  fold 
by  removing  it  two  places,  etc. ;  i.  e.  ten  units  of  the  first  order 

11.   One  hundred,  how  expressed?    Two  hundred?     13.   Other  uumhe  8,  how 
Md!     13.   What  is  the  place  of  a  figure?     What  does  the  flgui*  in  the 
eond  place?    Third?    14.   How  many,  and  what  value*  ~ 
liaa  a  figure?     15.   How  does  moving  a  flguru  towards  the  left  nflwet  its  vaiu«' 


10  NOTATION   AND   NLMEKAT: 

make  one  ten,  ten  tens  make  one  hundred,  ten  hundreds  make 
one  thousand,  and,  in  short,  ten  units  of  any  order  make  one  unit 
of  the  next  higher  order, 

10.  The  cipher,  when  used  with  other  figures,  fills  a  place 
that  would  otherwise  be  vacant ;  thus,  in  206  the  cipher  occupies 
the  place  of  tens,  because  there  are  no  te?is  expressed  in  the 
given  number. 

17.  The  figures  of  large  numbers,  for  convenience  in  read- 
ing, are  often  separated  by  commas  into  periods  or  group-. 

There  are  two  methods  of  numerating:  the  French  and  the 
English.  By  the  French  method  a  period  consists  of  three 
figures ;  by  the  English,  of  six.  The  French  method  is  most 
convenient,  and  principally  used  in  this  country. 

18.  By  the  French  Method  of  Numeration  the  first 
or  right-hand  period  contains  units,  tens,  and  hundreds,  and  is 
called  the  period  of  units  ;  the  second  period  contains  thousands, 
tens  of  thousands,  and  hundreds  of  thousands,  and  is  called  the 
period  of  thousands ;  etc.,  as  in  the  following 

FRENCH  NUMERATION  TABLE. 


i       il       1  i      i  ■,-      1 4      Is 


&%      GJ      «§      Ml 

smosS      OT  o  =3  c—  5 ~    • 

o^3       t-o'C       t^os       t-oc       t-oc       uoS 


13 
o 


hc  khc  wes  w#3  wSa  wSS  »Ss 

2  8,     7  6  9,     5  4  0,     7  0  6,     4-7  6,    0  0  1,     8  4  3. 


7th  period,    6th  period.  5th  period,  4th  jieriod,  3d  period,   2d  period,    1st  period, 
C^uiutillions. Quadrillions,  Trillions,     Billions,     Millions,    TIioumuius,       Uniw, 

10.  For  what  is  the  cipher  used?  17.  How  many  methods  of  numerating? 
What  are  they?  Which  is  generally  used  in  this  country?  1*.  Name  the  diff- 
erent periods  in  the  French  Numeration  Table.    Repeat  the  table. 


NOTATION    AND  TION. 


11 


10.     The    value  of    the    figures    in    thi>    table,    expp— ed    in 

.  i-  twenty-eight  quintillioo,  Mven  hundred  and  sixty-nine 

quadrillion,  live  bandied  and  forty  trillion,  seven  bandied  and 
mx  billion,  four  hundred  and  s<.vcnty-HX  million,  one  thousand, 
tight  bandied  and  fortj-tl 

1  10  iiKADixo  of  a  number  consists  of  two  distinct  pro  ■ 
nmting   the  order  of  the  placm,  beginning  >\t  the  light  hand  ;  thus, 
Mnu,  hnndreds,  etc.,  M  in  the  Numeration  Table;  and,  second,  n 
aim  of  the  figures,  btgfnaiBg  at  the  left,  as  above.     To  distil 
these  processes,  the  first  may  be  called  numcratiny,  and  the  second  reading, 
the  Dumber. 

20.  The  table  can  be  extended  to  any  number  of  places, 
adopting  a  new  name  for  each  succeeding  period.  The  periods 
above  quintillions  are  sextillions,  septillions,  octillions,  nonillions, 

lions,  undecillions,  duodecillions,  etc. 

21.  To  numerate  and  read  a  number  according  to 
the  French  method : 

BULK.  1.  Beginning  at  the  right,  numerate  and  point  off  the 
number  into  periods  of  three  figures  each. 

2.  Beginning  at  the  left,  read  each  period  separately,  giving 
the  name  of  each  period  except  that  of  units. 

Exercises  in  Numeration  by  the  French  Method. 


2S. 

Let  the  learner  read  the  following  numbers : 

1. 

24 

11. 

7,435,720,597  * 

2. 

357 

12. 

74,090,007.407 

3. 

4,649 

13. 

297,999,399,089 

4. 

95,679 

14. 

6,137,731,975,468 

5. 

549,:>  I  7 

15. 

i:».719,456,972,145 

6. 

5,745,328 

16. 

457,71  iU:;o,958,083 

7. 

52,073,712 

17. 

8,125,945,654,315,756 

8. 

213,967,184 

is. 

57,968,568,194,437,978 

9. 

4,674,925,178 

19. 

867,942,148,866,145,816 

10. 

48,404,876 

20. 

3,593,047,671,350,486,950 

19.    What  is  the  value  of  the  number  expressed  in  the  (able?     Heading  a  nutn- 
i-ists  of  how  many  processes?    What  are  they?    80.  What  are  the  names 
of  periods  above  Quintillions?    21.  Rule  for  numerating  and  reading  a  number 
by  the  Frenoh  method  r 


12  NOTATION    AND    N  [OK. 

23.    To  write  numbers  by  the  French  method  : 

Rcle.  1.  Beginning  at  the  left,  write  the  figures  belonging 
to  the  highest  period. 

2.  Write  the  figures  of  each  successive  period  in  their  order, 
filling  each  vacant  place  with  a  cipher. 

Exercises  in  French  Notation  and  Numeration. 

2-1:.  Let  the  learner  write  the  following  numbers  in  figures, 
and  read  them  by  the  French  method : 

1.  Two  units  of  the  third  order  and  five  of  the  fir 

Ans.  205. 

Note.  Since  no  figure  of  the  second  order  is  given,  acipha-  is  written  in 
the  second  place 

2.  Six  units  of  the  fourth  order,  three  of  the  second,  and 
eight  of  the  first.  An*,  <*.'>38. 

3.  One  unit  of  the  seventh  order,  three  of  the  sixth.  s«ven  of 
the  third,  and  two  of  the  second.  An  720. 

4.  Five  units  of  the  fifth  order  and  three  of  the  fourth. 

5.  Six  units  of  the  fourth  order  and  one  of  the  third. 

6.  Two  units  of  the  eighth  order  and  three  of  the  sixth. 

7.  Nine  units  of  the  ninth  order,  six  of  the  fifth,  one  of  the 
second,  and  three  of  the  first 

2«5.  Express  the  following  numbers  in  figures  by  the  French 
notation : 

1.  Three  hundred  and  fifty-six.  Ans.  356. 

2.  Six  hundred  and  fifty-three.  Ans.  653. 

3.  Five  hundred  and  sixty-three.  Ans.  563. 

4.  Three  hundred  and  sixty-five. 

5.  Six  hundred  and  fifty-one. 

6.  One  thousand,  six  hundred  and  fifty-one.         Ans.  1,651. 

7.  Forty-two  thousand,  five  hundred  and  fifty-four. 

8.  Eight  hundred  sixteen  thousand,  and  two  hundred. 

9.  Six  million,  one  hundred  four  thousand,  two  hundred  and 
seventy-six.  Ans.  6,104,270. 

S3.    Rule  for  wriUng  numbers  by  the  French  method? 


NOTATION    AND   NUMERATION.  13 

10.  Three  bandied  six  thousand,  five  hundred  and  two. 

11.  Nine  hundred  i'oriy-MX  million,  live  hundred  fourteen 
thousand,  nine  hundred  and  twenty-five. 

12.  Six  billion,  fifteen  million,  seven  thousand,  and  four  hun- 
dred Ans.  G,0 15,007,400. 

13.  Five  million,  six  hundred  fifty-one  thousand,  four  hundred 
and  six. 

14.  Seventy-four  million. 

15.  Sixty-three  million,  fourteen  thousand,  and  seven  hundred. 

26.  By  the  English  Method  of  Numeration,  the  first 
period  contains  units,  tens,  hundreds,  thousands,  tens  of  thou- 
sand-, and  hundreds  of  thousands,  and  is  called  the  period  of 
units;  the  second  period  contains  millions,  tens  of  millions,  hun- 
dreds of  millions,  thousands  of  millions,  tens  of  thousands  of 
millions,  and  hundreds  of  thousands  of  millions,  and  is  called 
the  period  of  millions  ;  etc.,  as  in  the  following 

ENGLISH  NUMERATION  TABLE. 

L 

*| 


la  „~  .        4 


o  |  g  i  ^       ©  | 


S  -*-  9  ©  «e  5?      T  •—   5  'o 

5  3  g  S  3  a       E  3  c  g  g  •- 

70647  G,      00184  3. 


lod,  "il  period,  1st  period, 

liillious,  Millions, 


20.    By  the  English  numeration  what  figures  arc  in  the  first  period?    Second 
period'    Third?     Repeat  the  tnb!e. 

2 


14  NOTATION   AND   NUMERATION. 

27.  The  value  of  the  figures  in  this  table,  is  twenty-right 
trillion,  seven  hundred  sixty-nine  thousand  five  hundred  ;md 
forty   billion,   seven    hundred    six    thousand   four   hundred    and 

;itv-six  million,  one  thousand  eight  hundred  and  fbrty-thn -e. 

28.  The  names  of  the  figures  and  their  values  are  the  same 
in  the  two  tables  for  the  first  nine  places  from  the  right,  After 
which  they  are  alike  in  value  but  different  in  name.  A  trillion 
by  the  English  method  is  much  more  than  by  the  French. 

29.  To  numerate  and  read  a  number  according  to 
the  English  method : 

BULK.  1.  Beginning  at  the  right,  numerate  and  point  off 
the  number  into  periods  of  six  figures  each, 

2.  Beginning  at  the  left,  read  each  period  separately,  giving 
the  name  of  each  period  except  that  of  units. 

Exercises  in  Numeration  by  the  English  Method. 

30.  Read  the  following  numbers  : 

87,658765,G47596 
95467,694164»745I 

47,G78600,709050,359G9l 

31.  To  write  numbers  by  the  English  method  : 
Kile.     1.  Beginning  at  the  left,  write  the  figures  belonging 

to  the  highest  period. 

2.  Write  the  figures  of  each  successive  period  in  their  ordert 
fitting  each  vacant  place  with  a  cipher. 

Exercises  in  English  Notation  and  Numeration. 

32.  Write  the  following,  and  read  by  the  English  method : 
1.   Five  units  of  the  eighth  order,  six  of  the  seventh,  two  of 

the  fourth,  and  one  of  the  third.  Ans.  56,002100. 

27.  What  number  is  expressed  by  the  table?  28.  Are  the  names  of  figure* 
alike  in  the  French  and  English  tables?  Their  values,  alike  or  unlike?  29.  Rule 
for  numerating  and  reading  a  number  by  the  English  method  ?  31.  Rule  for 
writing  a  number  by  the  English  method? 


1. 

4. 

2. 

8581 

5. 

3. 

7.17 

6 

NOTATION    AND    Nl  Mi:  RATI  ON.  1  6 

2.  Nino  units  of  the  fourteenth  order,  two  of  the  twelfth, 
three  of  the  eleventh,  >ix  of  the  eighth,  nine  of  the  sixth,  two 
Of  the  tilth,  and  three  of  the  fourth.     Am  90^30060,928000. 

3.  Two  units  of  the  ninth  order,  six  of  the  sixth,  one  of  the 
fifth,  two  of  the  third,  ;id.  ami  live  of  the  first. 

33.    Express  the  following  numbers  by  the  English  Notation : 

1.  Seventy-two  million,  six  hundred  thirteen  thousand  four 
hundred  and  forty-six.  Ans.  72,613446. 

2.  Five  hundred  seventeen  billion,  three  hundred  twenty-two 
thousand  one  hundred  fourteen  million,  eight  hundred  forty-one 
thousand  nine,  hundred  and  sixty-nine. 

3.  Two  hundred  and  ten  billion,  and  six  thousand. 

Note.     These  and  other  exercises  will  be  varied  and  extended  by  the 

teacher  as  circumstances  may  dictate. 

31.  The  Roman  Notation,  or  that  used  by  the  ancient 
Romans,  employs  seven  capital  letters  to  express  numbers,  viz.: 

I,       Y,       X,       L,  C,  D,  M. 

One,    Five,     Ten,     Fifty,    One  hundred,     Five  hundred,     One  thousand. 

All  other  numbers  may  be  expressed  by  combining  and  re- 
peating these  letters, 

3»>.  The  Roman  Notation  is  based  on  the  following  princi- 
ples : 

1st.  When  two  or  more  letters  of  equal  value  are  united,  or 
when  a  letter  of  less  value  follows  one  of  greater,  the  sum  of 
their  values  is  indicated;  thus,  XXX  stands  for  30,  LXV  for  65, 
CC  for  200,  MDCLXVII  for  1667. 

2d.  When  a  letter  of  less  value  is  placed  before  one  of  greater, 
the  difference  of  their  values  is  indicated ;  as,  IX  stands  for  9, 
XL  for  40,  XC  for  90. 

3d.  When  a  letter  of  less  value  stands  between  two  of  greater 
value,  the  less  is  to  he  taken  from  the  sum  of  the  other  two ;  as, 
XIV  stands  for  11,  XIX  for  19,  CXL  for  110. 

3».    How  many  and  what  characters  are  employed  in  the  Roman  Notation* 
a!ue  of  each?     35.   What  is  the  first  principle  in  Roman  Notation  t 
[I     Third? 


16 


NOTATION   AND  NUMERATION. 


4th.  A  letter  with  a  line  over  it  represents  a  number  one 
thousand  times  as  great  as  the  same  letter  without  the  line  ;  thus 
X  Staodfl  for  ten,  hut  X  stands  for  one  thousand  times  ten,  i.e. 
ten  thousand ;  M  stands  for  one  thousand,  but  M  for  one  thousand 
tvnes  one  thousand. 

TABLE   OF  ROMAN   NUMERALS. 


I 

1 

XVI 

16 

cccc 

400 

II 

2 

XVII 

17 

D 

500 

III 

3 

XVIII 

18 

DC 

GOO 

IV 

4 

XIX 

19 

DCCCC 

900 

V 

5 

XX 

20 

M 

1000 

VI 

6 

XXI 

21 

MI) 

15QQ 

VII 

7 

XXIV 

24 

MDC 

1G00 

VIII 

8 

XXV 

25 

MIX  LXV 

1GG5 

IX 

9 

XXIX 

29 

MDCCXLIX 

17-19 

X 

10 

XXX 

30 

MDCCCXV1 

1816 

XI 

11 

XL 

40 

HDCCCLXH 

1862 

XII 

12 

L 

50 

V 

5000 

XIII 

13 

LX 

L 

50000 

XIV 

14 

XC 

90 

C 

100000 

XV 

15 

c 

100 

M 

1000000 

Exercises  in  Roman  Notation. 


36.    Kx press  the  following  numbers  by  letters : 


1.  Twelve. 

2.  Eighteen. 


Ana.  XII. 
Ans.  XVIII. 


3.  Twenty-nine. 

4.  Ninety-nine. 

5.  Two  hundred  and  eijrhty-four. 

6.  One  thousand  four  hundred  and  forty-six. 

7.  One  thousand  six  hundred  and  forty-four. 

8.  The  present  year,  A.  D. . 

Note.  The  Roman  notation  is  very  inconvenient  for  Arithmetical  oper- 
ations, and  the  Roman  numerals  arc  now  seldom  used,  except  for  number- 
ing the  pages  of  a  preface,  the  divisions  of  a  discourse,  and  the  sections, 
chapters,  and  other  divisions  of  a  book. 


33.  What  is  the  fourth  principle  in  Roman  Notation?    36.  Are  Roman  nume- 
rals much  used  in  arithmetical  operations?    Why?    For  what  are  they  used? 


ADDITION.  17 

37.     Boridei  the  Arabic  and  the  Roman  figures,  there  are 

is  marks  used  to  indicate  that  certain  operations  are  to  be 

performed,  sneh,  e.  g.,  a>  the  sign  of  addition, -\-\  the  sign  of 

rfion,  — ;  etc.     These  rigQfl  will  be  given,  and  their  usea 

explained  when  their  aid  is  ne< 


ADDITION. 

38.  Addition  is  the  putting  together  of  two  or  more  num- 
bers of  the  same  kind,  to  find  their  sum  or  amount. 

The  sum  or  amount  of  two  or  more  numbers  is  a  number  which 
contains  the  same  number  of  units  as  the  two  or  more  numbers 
put  together;  thus,  7  is  the  sum  of  3  and  4,  because  there  are 
just  as  many  units  in  7  as  in  3  and  4  put  together ;  for  a  like 
reason  11  dags  is  the  sum  of  2  days,  4  days,  and  5  days. 

Ex.  1.  James  has  4  marbles,  John  has  5,  and  Henry  has  3 ; 
how  many  marbles  have  they  all  ? 

To  solve  this  example,  add  the  numbers  4,  5,  and  3  :  thus,  4 
and  5  are  9,  and  3  are  12;  therefore  James,  John,  and  Henry 
have  12  marbles,  Ans. 

2.  How  many  are  3  and  G  ?     6  and  3  ?    2  and  5  and  7  ? 

39.  A  Sign  is  a  mark  which  indicates  an  operation  to  be 
performed,  or  which  is  used  to  shorten  some  expression. 

40.  The  sign  of  dollars  is  written  thus,  $ ;  e.  g.  $2  repre- 
sents two  dollars  ;  $10,  ten  dollars,  etc. 

41.  The  sign  of  equality, =,  signifies  that  the  quantities  he- 
tweeo  which  it  stands  are  equal  to  each  other;  thus,  $1  =  100 
cents,  i.  e.  one  dollar  equals  one  hundred  cents. 

37.   What  characters  are  used  in  Arithmetic  besides  the  Arabic  and  Roman 

'     For  what ' 
as.    What  is  Addition?    Sam  or  amount?    39.    A  sign?    40.   Make  the  eign  of 
Jollari  on  the  black-board.    4  1.  Hake  the  sign  of  equality.    "What  does  it  meant 

2* 


18  ADDXTK 

42.  The  sign  of  addition,  4-,  called  plus,  denotes  that  the 
quantities  between  which  it  stands  are  to  be  added  together ; 
thus,  3-|-  2  =  5,  i.  e.  three  plus  two  equals  five,  or  three  and 
two  are  five. 

43.  Three  dots,  thus,  .*. ,  are  the  symbol  for  therefore,  hence,  or 
consequently ;  thus,  2  +  3  =  5,  and  3  +  2  =5, .-.  2  -f-  3  =  3  +  2, 
i.  e.  therefore  the  sum  of  2  and  3  is  equal  to  the  sum  of  3  and  2. 

Ex.  3.  William  paid  $4  for  a  pair  of  skates,  $3  for  a  sled,  and 
SI  for  a  knife  ;  what  did  he  pay  for  all  ? 

-S3  +  S1=$8,  Ans. 

4.  What  is  the  sum  of  SG  +  S3  ?      S">  -f  $2 -f  $8  ? 

5.  What  is  the  sum  of  4  +  6  +  2  +  3?      3  +  5  +  8  +  2? 

44.  To  add  when  the  numbers  are  large  and  the 
amount  of  each  column  is  less  than  10. 

6.  A  manufacturer  sold  125  yardi  of  cloth  to  one  merchant, 
342  to  another,  and  231  to  another;  how  many  yards  did  he  sell 
in  all?  Ans.  698. 

Haying   arranged    the*  numbers   so   that   units 
operation,    stand  under  Emits,  ten  ander  tens,  etc.,  add  the 

1  2  5       units  ;  thus    1    and   2   are  3,  and  5  are  8,  and 
3  4  2       set  the  result  under  the  column  of  units.     Then 

2  3  1        add  the  tens  ;  thus,  3  and  4  are  7,  and  2  are  9, 
Sum    fi9  8       set  ^0%vn   tne  result>  **d   so  proceed  till  all  tho 

columns  are  ad<! 

Ex.  7.         8.  9.  10. 

42  127  10  6  6  2  0  4 

14  3  3  4  1  3  4  1  2  4 1 8 

2  3  1  2  10  12  1  12  3  1 


Sum,  799  678        568  9848 
11.         12.          13.  14. 

2  0  00  1121       11200  100  0 

2  345  5127  To  413  2743 

1423  2  3  4  0      3  2  14  2  3  15  4 

3231  1 400      2  1034  100  1 


43.  Make  the  (sign  of  addition.     43.  Sign  for  therefore.    44.  How  are  nura 
b«jf  arranged  for  addition  ?     Which  column  is  added  first  ?   Ita  sum.  whei<  plaoed  I 


•   AUDIT  10 

15.  What  H  the  mm  of  1248,  2112,  and  1313  ?     Ans.  4668. 

16.  What  is  the  sum  of  2013,  1121,  1182,  an<l  1231  ? 

17.  A  gentleman  paid  sl2.">  fort  bone,  >231  for  a  chaise,  airj 
)t  a  harness  ;   what  did  lie  pay  for  all?  Ans.  $388. 

«M.    To  add  when  the  amount  of  any  column  is  10  o  * 
more. 

18.  Add  together  27,  93,  and  145.  Ans.  265, 

Having  arranged  the  numbers,  add  the  column 

operation.       of  units  ;  thus,  5  and  3  are  8,  and  7  are  15  units 

2  7  (  =  1  ten  and  5  units).     The  5  units  are  placed 

9  3  under  the  column  of  units,  and  the  1  ten  is  added 

1   1  5  to  the  column  of  tens  ;  thus,  1  and  4  are  .r>,  and 

Ans.  2  6  5  9  are  14,  and  2  are  16  tens  (=  1  hundred  and  6 

tens).     The  6  tens  are  set  under  the  tens,  and  the 

1  hundred  i>  added  to  the  1  hundred  in  the  third  column,  making 

2  hundreds  to  be  set  under  the  third  column. 


19. 

20. 

21. 

22. 

276 

748 

4681 

36487 

483 

249 

73  62 

10  4  6  2 

874 

838 

8428 

38420 

Ans.  16  3  3 

1835 

20471 

85369 

23. 

24. 

2,3. 

26. 

417 

246 

387  1 

34827 

819 

385 

1920 

5148 

234 

274 

4208 

97604 

846 

961 

3  186 

27 

72  1 

249 

8004 

86129 

Ans.  3  0  3  7 

27. 

28. 

29. 

80. 

46723 

4  6  28 

327 

3 

5  i 

943> 

56948 

784 

46 

4876 
and  426. 

98  643 

31.  Add  8467, 

82,  946,  18845, 

Ans.  18766. 

All  6  1287,  S  12,  8694,  32,  and  46872. 

8,  '.'7.  4T.82,  3800,  and  47289. 
84.  Add  884,  16842,  31,  87,  6294,  and  8274. 


20  ADDITION. 

46.  The  examples  already  given  embrace  all  the  prin- 
m  in  addition.     Hence,  to  add  numbers, 

Rule.  Write  the  numbers  in  order,  units  under  units,  tens 
under  tens,  etc.  Draw  a  line  beneath,  add  togtther  the  fg  tires  in 
the  units'  column,  and,  if  the  sum  be  less  then  ten,  set  it  under  that 
column  ;  but,  if  the.  sum  be  ten  or  more,  write  the  units  as  before, 
and  add  the  tens  to  the  next  column.  Thus  proceed  till  all  the 
columns  are  added. 

47.  Proof.  The  usual  mode  of  proof  is  to  begin  at  the  top 
and  add  downward.  If  the  work  is  right,  the  two  sums  will  be 
alike. 

n:  I,  By  this  process,  we  combine  the  figures  differently,  and  hence 
shall  probably  detect  any  mistake  which  may  have  been  made  in  adding 
ujimird. 

ILLUSTRATION.  Tn     ■**«     *P"***  WG  «*!   2  ««"*    C  ■* 

Ex  35  8,  and  7  are  15,  and 4  are  19, etc;  bat 

in  adding  downward,  we  saw  1  and  7  an 

11,  and  6  are   17,  and  2  are  19,  etc.,  thus 


obtaining  the  same  result,  but  by  different 


4  8  2  9  7 

nig**  ***** 

If  we  do  not  obtain  the  same  result  by 


Sum,     2  4  9  5  7  9         the  two   methods,   one   operation  or  the 
Proof,   2  4  9  5  7  9         other   is    wrong,    perhapfl    both,  and  the 
work  must  be  carefully  performed  again. 

te  2.  Tn  adding  it  is  not  desirable  to  name  the  figures  that  we  add ;  thus, 
in  example  35,  instead  of  saying  2  and  6  are  8,  and  7  are  15,  and  4  are  19,  it 

rter,  and  therefore  better,  to  say  2,  8,  15,  19  ;  setting  down  the  9,  say  1, 
6,  10,  19,27 

36.  What  is  the  sum  of  8432,  42698,  34,  1892,  70068,  5142, 
and  68742?  Ana,  197008. 

37.  What  is  the  sum  of  2468,  13579,  276,  and  4_ 

38.  What  is  the  sum  of  3406,  872,  6.V11,  2,  and  17  ? 

39.  What  b  the  sum  of  3910,  4,  876,  27,  and  89462  ? 

46.  If  the  amount  of  any  column  is  ten  or  more,  where  is  the  right-hand  figure 
of  the  amount  writteu  ?  What  is  done  with  the  left-hand  figure?  Repeat  the  rule 
for  Addition.  47.  How  is  Addition  proved?  Why  not  add  upward  a  second 
time  ?    Is  it  desirable  to  name  the  figures  as  we  add  them? 


ADDITION. 

21 

Ex.  40. 

n. 

42. 

43. 

51000 

20404 

21153 

31201 

1  1  G08 

44346 

25000 

22222 

880  80 

•  40 

15  000 

6  6  6  6  6 

4  9  l  8  I 

9  0000 

5  5  5  5  5 

5  5  5  5  5 

l  2  B 

95000 

5  4  4  1  5 

3o 

-2643 

•II.   HOW  many  are  876 -f  9287  +  69842  +  7700  ? 

Ans.  87705. 
45.  How  many  are  3G904  +  21G  -f  8942  +  47  ? 
Bow  many  are  18  +  4  +  7G984  -f  327  +  1  1  ? 

47.  84G  +  972  +  84  -f  300  =  how  many  ?         Ans.  2202. 

48.  2468  +  98G7  +  37428  +  278  =  how  many  ? 

49.  3004  +  6094  -j-  87642  -f  36  =  ?  Ans.  9G77G. 
„  168  +  13579  +  100  +  6042  +  187  +  19  =? 

51.  Add  four  hundred  and  sixty-two;  three  thousand  two 
hundred  and  fourteen  ;  seventy-nine  thousand  six  hundred  and 
hfty-nine;  and  two  hundred  and  eighty-four.  Ans.  83619. 

Add  four  hundred  and  iif'ly-six  ;  eight  thousand,  four  hun- 
dred and  BeVenty-tWO j  fifteen  thousand,  seven  hundred  and 
twenty-one  ;  forty-three  million,  seven  hundred  and  thirty-three 
thousand,  eight  hundred  and  fifty-nine;  and  ten. 

53.  The   population  of  England  in   1851  was  16921888;  of 
and,  2888742;   of  Wales,  1005721;   of  Ireland,  G515794. 
What  was  the  population  of  Great  Britain  and  Ireland? 

B  I.  England  and  Wales  contain  about  55100  square  miles; 
Scotland  29600 j  and  Ireland,  32000  ;  what  is  the  area  of  the 
British  Islands.?  Ans.  116700  square  miles. 

l)y  the  census  of   1860,  the  number  of  inhabitants  of 
Maine,  was  628276;  of  New  Hampshire,  826072;  of  Vermont, 
815116;  of  Massachusetts,  1231065;  of  Rhode  Island,  174621  ; 
of  Connecticut,  4G0151  ;  what  was  the  population  of  New  Eng- 
land? Ans.  3135301. 
5G.  The  area  of  Maine  is  35000  square  miles;  N.  II.,  8030; 
•no;  Mass.,  7250;  R  I.,  1200;  Ct,  4750.    What  fa  the 
I  &  New  England  ? 


22  ADDITION. 

*<7.  In  1850  the  population  of  Maine  was  583169;  of  New 
Hampshire,  317076;   of  Vermont,  514120;   of  nfirmrnnni 

14;    of  Bhodfl   Island,   117  Connecticut,  370702; 

what  was  the  population  of  these  *ix  Stales  in 

58.  A  merchant,  commencing  business,  h;i<l  in  cash,  $4376  ; 
goods  worth  $3780;  hank  Itock  worth  $2700;  and  other  prop- 
erty valie  ;.»f>.     In  a  year  be  gained  $2475;  irnsd 

he  worth  at  the  end  of  the  year? 

59.  In  one  year  a  l  id  a  pair  of  oxen  for  $125,  two 
cows  for  $75,  three  swine  for  $'.»  p  for  $120,  and  a 
bone  for  $156  ;  what  did  lie  receive  for  all  ? 

CO.  On  Monday, a  merchant  .-old  goods  for  $:V>7,  on  Tuesday, 

rhnrsday,  lor  S3 18,  on 
Friday,  I  I    on    Saturday   for    $316;   what  was   the 

value  of  the  goodfl  .-old  during  the  week  ? 

61.  In    1850  the  population  of  New  York  :  of 

Philadelphia,  840045 ;  of  Baltimore,  169054;  of  IV  5881; 

of  New  Orlea  sad  of  Cincinnati,  11543$;  whal 

number  of  inhabitants  in  these  six  cities  in  1850? 
.'.   In  the  middle  of  the  nim •:■  the  population  of 

l  ;  of  Paris,  1053897  j  tan* 

tinople,  786990  ;  c  ~  ;  of  Vienna.  4771 

of  Berlin,  441931  ;  and  of  Nap!  tbe  popu> 

63.  In  1851  ion  of  the  (Jailed  Stat-  ,i>out 

1876;  of  Britain  and  116181427882145;  of  Fr; 

-3170;    of  R.  88000;    and  of  Aosftri 

what  wai  the  population  of  these  five  count] 

,.  The  population  of  North  America  i>  about  30257819;  of 
South  America,  18878188;  of  Europe,  265368216;  of  A 
of  Africa,  61688770,  and  of  Oceanica,  23444- 
what  is  about  the  population  of  the  globe  ?     Ans.  1038803745. 

of  the  American  army  for  li 
commencing  in  1812.  was  $12187048,  $19906362,  $20008306, 
;<>0,  and  #16475412  ;  what  was  the  cost  for  five 

\  to  D  $4082,  to  E  $207,  to  F  $54, 
imd  to  G  $1353;  how  much  does  he  owe? 


ABDRXO 

I 

68. 

69. 

70. 

'98 

5  9 

28738 

. 

5  8  7. 

7  9  819 

9  03 

a  i 

0  1 

l  8  5  8  2 

7/.  | 

55555 

4  l  2 

93977 

27579 

1  2  6  7  7 

505  04 

11111 

2  !  7  6  1 

4  7  80  1 

56  6 

88888 

I  1  9  1  1 

8  7  0  1  8 

8  47  69 

5  5  i 

;  6  i 

2525] 

I  6  9  0 

;98 

2  12  7  4 

1  2  1  6  $ 

6  i 

i  90 

28 

5  I  0  00 

i  i 

98G95 

72869 

•8  78 

2  5 

D  6  5  6  1 

27  121 

4  0  5  0  2 

2  7  2  0  8 

90.^ 

46^ 

1  6  5 

92672 

6  2  12  8 

2  7 

•2177  8 

j  6  7 

74279 

6  l  • 

2  5  1  2  7 

7  6  1  5  2 

2  4  7  2  5 

5  2 

9  l 

267 

76592 

27248 

47214 

73017 

15172 

47510 

23 


71.  In  January  there  are  31  days,  in  February  28,  in  March 
31,  in  April  30,  in  May  31,  in  June  30,  in  July  31,  in  August 
31,  in  September  30,  in  October  31,  in  November  30,  and  in 
mber  31  ;  how  many  days  are  there  in  a  year  ? 
7  J.  A  gardener  has  3476  apple  trees,  847G  pear  trees,  5684 
peach  trees,  1845  plum  trues,  4680  quinee  trees,  and  9  187  orna- 
mental trees;  how  many  trees  are  there  in  his  nursery? 

73.  The  first  of  three  numbers  is  4768,  the  second  is  8942, 
and  the  third  is  as  much  as  the  other  two  ;  what  is  the  sum  of 
the  three  numbers? 

7  1.   I  have  $376  in  one  bank,  $1078  in  another,  and  in  anoth- 
much  as  in  both  of  these  ;  how  much  money  have  I  in  the 
three  banks  ? 

.   An  army  consists  of  276450    infantry,    14875   cavalry, 
i  artillery  men,  and  127462  riflemen;  what  is  the  number 
of  men  in  the  army  ? 

A  carpenter  engaged  to  build    1  houses,  the  first  I 
eood   i'nv  e   third   t  .   and   the  fourth  for 

$12469  ;   what  shall  he  receive  for  the  lour  bous< 


24  SUBTRACTION. 


SUBTRACTION. 

48.  Subtraction  is  taking  a  less  number  from  a  greater 
number  of  the  same  kind,  to  find  their  difference. 

The  greater  number  is  called  the  minuend;  the  less  nun 
is  called  the  subtrahend  ;  and  the  result  is  called  the  differ- 
B  or  remainder. 
Ex.  1.  Arthur  had  7  apple-,  hut  be  has  given  4  of  them  to 
Mary;  how  many  apple.-  has  lie  : 

18,  3j  because  4  apple-  taken  from  7  apples  leave  3  apples. 

2.  John  having  17  marble-,  lost  7  of  them  ;  how  many  had 
he  left  ? 

4:9.  The  sign  of  subtraction, — ,  called  minus,  signifies  that 
the  number  after  it  is  to  be  taken  from  the  number  before  it; 
thus,  7  —  4  =  3,  i.  e.  seven  minus  four,  or  seven  diminished  by 
four,  equals  th: 

3.  How  many  are  10 —  6?  Ans.  4. 

4.  How  many  are  12  — 8?     12  —  4?     1G  — C? 

I  e.     When  the  numbers  are  small,  the  subtraction  is  ren<l:h 
in  the  mind;  but  when  they  are  large,  the  work  is  more  easily  done  by  writ- 
ing the  figures,  as  in  the  following  examples. 

50.    To  subtract  when  no  figure  in  the  subtrahend  is 

greater  than  the  corresponding  figure  in  the  minuend. 

5.  From  796  take  582. 

operation.  This  example  is  solved  by  taking  the 

Minuend,  7  9  6         2  units  from  6  units,  8  ten-  from  i' 

Subtrahend,      5  8  2         and  5  hundreds  from  7  hundreds,  giving 
aainder,       2T1         -u  lbr  the  remainder. 

6.  7.  8.  9. 

Minuend,       469  5642  9874  8  0  78 

Subtrahend,  327  4130  3623  S03J 

Remainder,    142  1512  6251  5021 

48.  What  is  Subtraction?  Minuend?  Subtrahend?  Remainder?  49.  Make 
the  sign  of  subtraction.  Its  meaning?  How  do  we  subtract  when  the  number* 
are  small  ?    How  when  they  are  large? 


SUBTRACTION-. 

From 
Take 

10. 
2741 
13  0  1 

11.                   12. 
5  4                   G  4  0  8 
1350             3207 

13. 
8420 
3110 

2.3 


Ant,  14  4  0 

14.  A  fanner  bought  a  farm  for  $1875  and  sold  it  again  for 
S  ;  bow  much  did  be  lose  by  the  transactions  ?  An*.  $1412. 

15.  By  the   census  of  18G0,  the  population  of  Maine   was 

id  that  of  New  Hampshire  was  826072;  how  many 
more  people  were  there  m  Maim;  than  in  New  Hampshire? 

16.  If  I  borrow  $4687  and  afterwards  pay  $2423,  how  much 
do  I  still  owe  ? 

•51.  To  subtract  when  any  figure  in  the  minuend  is 
less  than  the  corresponding  figure  in  the  subtrahend. 

17.  From  483  take  257. 

There  are  two  methods  of  explaining 

OPERATION.  ..  r  ° 

this  operation : 
Minuend,  4  8  3  Uu   As  we  cannot  take  7  units  from 

Subtrahend,       2_o_7         3  unUs  Qne  of  the  8  tem  fa  ])l]t  with  (he 

Remainder,        2  2  6  3  units,  making  18    unit-,  and   then,  7 

units  from  13  units  leave  G  units.  Now 
as  one  of  the  8  tent  lias  been  pot  with  the  3  writs,  only  7  tens 
remain  in  the  minuend,  and  5  tens  from  7  tens  leave  two  tens, 
and,  finally,  2  hundreds  from  4  hundreds  leave  2  hundreds;  .-. 
the  entire  remainder  is  226. 

2d.  Instead  of  taking  away  1  of  the  8  tens  in  the  minuend,  we 
may  aid  1  ten  to  the  5  te&l  in  the  subtrahend,  and  then  take  the 
sum  (G  tens)  from  the  8  tens,  since  the  result  is  2  tens  by  either 

. 

second  mode  depend-;  on  the  principle,  (hat,  if  two  mnn- 
bers  are  crjmilh/  increased,  (lie  difference  between  them  remains 
unchanged  :  thus,  the  difference  between  9  and  4  i-  5,  and,  if  10 
both  '.»  and  4.  making  19  and  II.  the  difference  still  is 
5.  NOW,  in  solving  Ex.  17  by  the  second  method,  we  add  10 
ttni'/s  to  tiie  minuend  and  1  ten  (the  same  as  10  units)  to  the 
sukraJicnd,  and  .-.  find  the  same  remainder  us  by  the  first  method. 

51.  How  many  methods  of  subtracting  when  a  figure  of  the  minuend  is  Jess  than 
the  one  under  it?  What  is  the  first  method?  Second?  The  second  depends  on 
what  principle*  By  the  second  method,  is  Ote  same  number  added  to  minuend  and 
Biibtrahtncl '     flbwl 

3 


26 


SUBTRACTION. 


52.  The  preceding  examples  illustrate  all  the  prin- 
ciples in  subtraction.     Hence,  to  perform  subtraction, 

BULB.  I.  Write  the  less  number  under  the  greater,  units, 
under  tinils,  tens  wider  tens,  etc.,  and  draw  a  line  benefit//. 

2.  Beginning  at  the  right  hand,  take  each  figure  of  the  subtra- 
hend from  the  figure  above  it,  and  set  the  remainder  under  the 
line. 

3.  If  any  figure  in  the  subtrahend  is  greater  than  the  figure 
above  it,  add  ten  to  the  upper  figure  and  take  the  lower  figure 
from  the  SUM  ;  set  down  the  remainder  and,  considering  the  next 
figure  in  the  minuend  one  less,  or  the  next  figure  in  the  subtra- 
hend one  greater,  proceed  as  before. 

•7&.  Proof.  Add  the  subtrahend  and  the  remainder  to 
gcther,  and  the  sum  should  be  the  minuend. 

Note  1.  This  proof  rests  upon  the  self-evident  truth,  that  the  whole  of  a 
tiling  is  equal  to  the  sum  of  all  its  parts ;  thus  the  minuend  is  separated  into  the 
two  parts,  subtrahend  and  remainder  ;  hetwe  the  sum  of  those  parts  must  be  the 
minuend. 

Ex.  18. 

Minuend, 
Subtrahend, 

Remainder, 

Proof, 


Minuend, 
Subtrahend, 

Rrmainder, 

Proof, 


68745 
26854 

41891 

68745 

10. 

9875 

265 

9610 


As  the  sum  of  the  subtrahend  and 
remainder  is  the  minuend,  the  work 
is  supposed  to  be  right. 


9875 


20. 
532769 
278493 

254276 
532769 


21. 
5784268 
329641 6 


22. 
From      4  6  8  7  2  4 
Take       259782 

Ans.   2  0  8  9  4  2 


23. 

5  4  0  6  8  7  2 
2304798 


24. 
9846237 
9468714 


00,    The  rule  for  Subtraction? 
principle  does  this  proof  re*t  I 


53.    How  is  Subtraction  proved?     On  what 


:  FACTION*.  27 

2G.  27. 

Fn.m   2487  69       8S7G04«      7777777 
Take   243  627       2  9  6  0  0  4  0      5666669 

Here  we  cannot  take  8  from  2, 
Ex.  28.  nor  can  we  borrow  from  the  tens' 

^  ^f\  ^V  Pulce>  as  tnnt  place  's  occupied  l>v 
Minuend,  0  0  2  0 ;  but  we  can  borrow  one  of  the 
Subtrahend,    4     3     8  G  hundreds  an(1   geparate  the  one 

Kiinder,     16     4         hundred  into  9  tens  and  10  units  ; 
then,    putting   the    0    tens    in    the 
place  of  tens  and  adding  the  10  units  to  the  2  units,  we  can  sub- 
tract 8  from  12,  3  from  9,  and  4  from  5. 

Note  2.  This  process  will  probably  bo  more  readily  understood  by  the 
young  learner  than  the  second  method  given  in  the  rule,  though  the  latter, 
being  thought  more  convenient,  is  usually  adopted. 

29.  30.  31. 

From  8702  4003  870000 

Take    2465  1876  324872 


32.  From  804  take  567.  Ans.  237. 

33.  From  4687  take  2398. 

34.  From  87062  take  36981. 

35.  Subtract  2437  from  8064.  Ans.  5627. 

36.  Subtract  160874  from  4769872. 

37.  Subtract  3768942  from  7000000. 

38.  Take  87406  from  95472.  Ans.  8066. 

39.  Take  2704698  from  8749206. 

40.  How  many  are  3642  less  1468?  Ans.  2174. 

41.  How  many  are  87649  less  24065  ? 

42.  8749  —  3684  =  how  many  ?  Ans.  5065. 

43.  7248  —  2943  =  how  many  ? 

44.  The   difference    between   two   numbers   is   365  and  ^he 
greater  number  is  876  ;  what  is  the  less?  Ans.  511. 

45.  What  number  added  to  3876  will  give  7469  ? 

46.  What  number  taken  from  8742  leaves  3748  ? 

53.  What  is  there  peculiar  iu  Ex.  28?    Explain  the  process. 


28  SUBTRACTION. 

47.  The  sum  of  two  numbers  is  8629,  and  the  less  of  the  two 
numbers  is  2G89  ;  what  is  the  greater?  Ans.  5940. 

48.  The  sum  of  two  numbers  is  8426,  and  the  greater  is 
7162  ;  what  if  the  less  ? 

49.  From  fourteen  million,  eight  hundred  and  sixty-two  thou- 
sand, three  hundred  and  twenty-five,  take  six  million,  six  hundred 
and  eighty-six  thousand,  two  hundred  and  fourteen. 

Ans.  8176111. 

50.  From  seven  hundred  and  thirty-three  thousand,  six  hun- 
dred and  fifty-four,  take  two  hundred  and  twenty-seven  thousand, 
five  hundred  and  fifteen. 

51.  How  many  years  from  the  discovery  of  America  by  Co- 
lumbus in  1492  to  the  birth  of  Washington  in  1782  \ 

59.  How  many  years  have  elapsed  since  the  discovery  of 
America  in  1492  ? 

53.  By  the  06BSM  of  1860,  the  number  of  inhabitants  in 
Massachusetts  was  1231065,  and  the  number  in  Vermont  was 
315116  ;  how  many  more  in  Massachusetts  than  in  Vermont? 

54.  The  population  of  the  United  Stat.-  wm  23191876  in 
1850,  and  17063353  in  1840;  what  was  the  increase  in  ten 
years  ? 

65.  The  area  of  New  England  is  64230  square  miles  and  the 
area  of  Maine  is  35000  square  miles;  what  is  the  area  of  the 
other  five  New  England  St:r 

56.  About  56619608  bushels  of  corn  were  raised  in  Ohio  in 
1850,  and  73436690  bushels  in  1888 ;  what  was  the  increase? 

57.  Bought  a  paper  mill  for  $15475,  and  sold  it  for  $17925  ; 
what  did  I  gain  ? 

58.  How  many  are  876942  —  468279  ? 

59.  How  many  are  742006  —  387429  ? 
GO.  How  many  are  820654  —  260408  ? 

61.  Washington  was  born  in  1732  and  died  in  1799  ;  at  what 
age  did  he  die  ? 

62.  A  merchant  sold  goods  to  the  amount  of  $4276,  and 
thereby  gained  $1142  ;  what  did  the  goods  cost  him? 

63.  A  farm  was  sold  for  $3462,  which  was  $876  more  than  it 
cost ;  what  did  it  cost  ? 


ADDITION    AND    SUBTRACTION.  29 

84  The  distance  from  the  earth  to  the  sun  is  about  95000000 
miles;  the  distance  to  the  DMOO  is  about  210000  miles.  How 
much  further  to  the  sun  th:in  to  the  moon  ? 

65.  Methuselah  died  at  the  age  of  969  years,  and  Washington 
at  67  ;  what  was  tin*  ditlerenee  of  their  ages? 

C6.  Mr.  Hale,  owing  a  debt  of  $3762,  paid  $24S6 ;  how  much 
remained  unpaid  ? 

LMPLES  in  Addition  and  Subtraction. 

1.  From  the  sum  of  7G  and  92  take  14.  Ans.  154. 

2.  From  the  sum  of  the  three  numbers,  876,  493,  and  91G, 
take  the  sum  of  842  and  397.  Ans.  1046. 

3.  I  owe  3  notes,  whose  sum  is  $600.  One  of  these  notes  is 
for  $150,  another  for  $200 ;  for  what  is  the  third  one  ? 

•1.  My  real  estate  is  valued  at  $4500  and  my  personal  prop- 
erty at  $2596.  I  owe  to  A  $600,  to  B  $1358,  and  to  C  $318  ; 
what  am  I  worth?  ■  Ans.  $4820. 

5.  Bought  a  barrel  of  flour  for  $9,  four  yards  of  cloth  for  $2, 
and  8  pounds  of  sugar  for  $1.  In  payment  I  gave  a  ten  and  a 
five  dollar  bill ;  what  change  shall  the  merchant  return  to  me  ? 

6.  Mr.  Fox,  owning  3762  acres  of  land,  gave  563  acres  to  his 
oldest  son,  and  072  acres  to  his  youngest  son;  how  many  acres 
had  be  remaining? 

7.  The  area  of  Maine  is  35000  square  miles ;  N.  IT.,  8030 ; 
B000 ;  Mass.,  7250  ;  R.  I.,  1200  ;  Ct.,  4750.     Which  is  the 

greater,  Maine  or  the  rest  of  N.  E.  ?     How  much  ? 

8.  Gave  my  note  for  $8465.  Paid  $1300  at  one  time,  and 
$575  at  another;  how  much  do  I  still  owe?  Ans.  $1590. 

9.  Mr.  T.,  opening  an  account  at  the  Andover  Bank,  deposited 
$187  on  Monday.   $362  on  Tuesday,  $580  on  Thursday,  and 

OH  Friday.     On  Tuesday  he  withdrew  $67,  on  Wednesday 
OO   Friday  $350,  and  on  Saturday   $125;  how  much  re- 
mained on  deposit  at  the  close  of  the  week?  Ans.  $1049. 

10.  A  traveler  who  was  875  miles  from  home,  traveled  to- 
ward home  141  miles  on  Monday,  127  miles  on  Tuesday,  156 
miles  on  Wednesday,  and  157  miles  on  Thursday;  how  far  from 
home  was  he  on  Friday  morning  ? 

3* 


30  MULTIPLICATION. 

11.  From  the  discovery  of  America  by  Columbus  in  1492,  to 
the  settlement  Of  Jam^town  in  1G07,  was  115  years,  from  the 
K -ill. -incut  of  Jamestown  to  the  Declaration  of  Independence  in 
177*'..  was  1G9  years,  and  from  the  Declaration  of  Independence 
to  the  present  time  (1862)  is  86  years.  Methuselah  died  at  the 
age  of  969  years;  how  much  longer  did  he  live  than  from  the 
discovery  of  America  to  the  year  1862  ?  j 

1 2.  Four  men,  A,  B,  C,  and  D,  commencing  business  together, 
famished  money  as  follows:  A?  18475]  B,  $8475]  C,  $2850| 
and  D,  $4500.  At  the  end  of  a  year  they  closed  business,  hav- 
ing lost  §o'22o  ;  how  much  money  had  they  to  divide  between 
them? 


MULTIPLICATION. 

54.  Multiplication  is  a  short  method  of  adding  equal 
numbers ;  that  is,  multiplication  is  a  short  method  of  finding  the 
sum  of  the  repetitions  of  a  number. 

Or,  Multiplication  i-  a  short  method  of  finding  how  many 
units  there  are  in  any  number  of  times  a  giv<n  i. umber. 
Mi  i.nri.K  and  i>  the  number  to  be  repeated 

The  Multiplikr  is  the  number  which  shows  how  many  times 
the  multiplicand  is  to  be  taken. 

The  Product  is  the  sum  of  the  repetitions,  or  the  result  of  the 
multiplication. 

The  Multiplicand  and  Multiplier  are  called  Factors. 

Ex.  1.  There  are  7  days  in  1  week;  how  many  days  in  4 
wee' 

This  example  may  be  solved  by  addition  ;  thus,  7  — |—  7  — [-  7  — |—  7 
r=  *J3 ;  or  more  briefly,  by  multiplication ;  thus,  4  times  7   are 
Yns. 

54.  "What  is  Multiplication?  Another  definition?  What  is  the  Multiplicand? 
Multiplier?    Troduct?    What  are  the  Multiplicand  and  Multiplier  called? 


MCLTIPLICYTION. 


31 


•!«>.    Tlie  pupil,  before  advancing  further,  should  learn  the 
MULTIPLICATION   TABLE. 


following 


Once 

Twice 

Three  limes 

Four 

times 

Five  times 

Six  times 

1 

b     l 

1  a: 

1  are    3 

1  are    4 

1  are    5 

1  are    6 

2 

I 

1 

•1 

2 

6 

2 

8 

2 

10 

2 

12 

3 

6 

9 

3 

12 

3 

]:, 

18 

4 

4 

1 

8 

4 

19 

■1 

16 

1 

20 

1 

24 

5 

0 

5 

10 

5 

15 

5 

20 

0 

25 

S 

30 

6 

a 

19 

6 

18 

8 

24 

6 

30 

8 

36 

7 

7 

7 

1  l 

7 

2] 

7 

28 

7 

35 

7 

42 

8 

8 

8 

16 

8 

84 

8 

32 

8 

40 

48 

9 

9 

9 

18 

9 

27 

9 

86 

9 

45 

9 

54 

10 

10 

10 

20 

10 

30 

10 

40 

10 

50 

K) 

60 

11 

11 

11 

89 

n 

33 

11 

44 

11 

55 

11 

66 

12 

12 

19 

u 

12 

36 

12 

48 

12 

60 

i12 

72 

Seven  times 

Eight 

times 

Bm  times 

Ten  times 

Eleven  times 

Twelve  times 

1  are    7 

1  are    8 

1  are    9 

1  are  10 

1  are  1 1 

1  are  1 2 

2 

14 

2 

16 

2 

18 

2 

20 

2 

22 

2 

21 

3 

21 

3 

24 

3 

27 

3 

30 

3 

33 

3 

36 

4 

88 

4 

82 

4 

86 

4 

40 

1 

44 

4 

48 

5 

35 

5 

40 

5 

45 

r> 

50 

5 

55 

5 

60 

G 

49 

6 

48 

6 

54 

6 

60 

6 

66 

6 

72 

7 

7 

T>6 

7 

C3 

7 

70 

7 

77 

7 

84 

8 

5G 

8 

64 

8 

72 

8 

80 

8 

88 

8 

96 

9 

G3 

9 

9 

81 

9 

90 

9 

99 

9 

108 

10 

To 

10 

80 

10 

90 

10 

100 

10 

110 

10 

120 

11 

77 

11 

88 

11 

11 

Hit 

11 

121 

11 

132 

12 

84 

19 

96 

12 

108 

12 

120 

12 

132 

12 

144 
j 

2.  How  many  are  8  times  3?  3  times  8?  6  times  4? 
4  times  6?     7  times  7  ?     5  times  9? 

3.  How  many  are  9  times  7  ?  9  times  11?  8  times  6? 
6  times  12?     12  timet  6?     9  times  8? 

•1.  If  I  deposit  SIO  a  month  in  a  MVlDgS  hank,  how  many 
dollar*  shall  I  deposit  in  4  months?  In  7  months?  In  o  months? 
In  12  months? 


82  MULTIPLICATION. 

5.  When  wood  is  worth  SG  a  cord,  what  shall  J  pay  for  3  cord~? 
5  cords  ?     8  cords  ?     11  cord?  ? 

G.  In  one  year  there  are  12  months,  how  many  months  in  2 
years?     4  years?     7  years?     12  year-? 

7.  If  I  study  11  hours  in  a  day,  how  many  hours  shall  I 
study  in  0  days?     5  days?.   8  days?     11  days? 

•56.    To  multiply  by  a  single  figure. 

8.  In  one  bushel  are  32  quarts ;  how  many  quarts  in  G 
bushels? 

by  addition-.  bt  nuLTi plicatiox.       In    G   bushel*   there  are, 

3  2  3  2              lently,  6  times  m  many 

3  2  6        quarts  as  in  1  bushel,  and 

3  2  -p     ,           i~o~o          tne  numoer  °f  quarts   in   G 

3  2  lrcKluct>    '•            bushel*    may   be    obtained 

3  2  by  adtlinfj,  as  in  the  margin;  or,  more  briefly, 

3  2  by  multiplying ;  thus,  G   times   2    units   are    12 

S„m  i  o  o         units  =  1   ten   and   2    units;  write   the   2  units 
UIIJ,  X    J   —  .  . 

in   umt>    place,  and   then    say  0  tlDQ  are 

18  tens,  which,  increased  by  the  1  ten  previously  obtained, 
make  19  tens  =  l  hundred  and  9  tens,  and  these,  written  in  the 
place  of  hundreds  and  tens  respectively,  give  the  true  product. 
Hen 

RULE.  Writs  (he  multiplier  under  the  multiplicand,  and  draw 
a  line  beneath  ;  multiply  the  units  of  the  multijilicand.  set  the  iinits 
of  the  product  under  the  multiplier,  and  add  the  QP»y,  to  the 

product  of  the  tens,  and  so  proceed. 

9.                       10.                          11. 
Multiplicand,      427                  1347                   1064 
Multiplier,  2  5  8 

Product,  SJ1  6  7  3  5  85  12 

11         13.  14.  15. 

8423      5436      26493      76489 
7         9  3  4 


5  8  9  G  1 


5G.  Which  flgure  of  the  Multiplicand  is  multiplied  first?    Where  are  the  uuits 
of  the  product  written?    What  is  done  with  the  tens?    Eepeat  the  ruJe. 


Ml  I/TII'LK'ATIOX.  33 

16,  17.  18. 

141  4787243  3424270 

6  9  7 


2  16  2  5  2  88969890 

•57.    To  multiply  by  two  or  more  figures. 

19.   How  many  (marts  in  46  hu.-dn-l-? 

operation-.  First  multiply  by  6,m  though  G  w<>rn 

Multiplicand,  3  2  the  only  figure  in  the  multiplier  ;  then 

Multiplier,       4  6         multiply  by  4,  and  set  the  first  figure  of 

-  q  a         tlns  product  in  the  place  of  tens  ;  for 

.  2  g  multiplying  by  the  4  tens  is  the  same  as 

multiplying  by  40,  and  40  times  2  units 

Product,     14  7  2         are  80  writs  =  8  tens;  i.  e.  the  product 
of  units  by  tens  is  tens.     Having  multi- 
plied by  each  figure  in  the  multiplier,  the  sum  of  the  partial 
products  will  be  the  true  product. 

Note.  So  much  of  the  product  as  is  obtained  by  multip/ying  the  whole 
multiplicand  by  one  jiyure  of  the  multiplier  is  called  O partial  product ;  thu>, 
in  the  19th  example,  192  is  the  first  partial  product  and  128  tens  is  the 
second. 

•58.  Similar  reasoning  applies  however  many  figures  there 
may  be  in  the  multiplier.     Hence, 

Rule.  1.  Set  the  multiplier  under  the  multiplicand  and  draw 
a  line  beneath. 

2.  Beginning  at  the  right  hand  of  the  multiplicand,  multiply 
the  multiplicand  by  each  figure  in  the  multiplier,  setting  the  first 

figure  of  each  partial  product  directly  under  the  figure  of  the 
multiplier  which  produces  it. 

3.  The  SUM  of  these  partial  products  will  be  the  true,  product. 

•59.    Proof.     Multiply  the  multiplier  by  the  multiplicand, 

and,  if  correct,  the  result  will  be  like  the  first  product. 

Hon.  This  proof  rests  on  the  principle,  that  the  order  of  the  factors  is 
immaterial;  thus,  3X4  =  4X3  ;  5X2X7  =  2X"X5. 

57.  Which  figure  of  the  multiplier  is  first  employed?  Where  is  the  first  figure 
of  etch  partial  product  written?  What  it  a  partial  product  ?  5R.  Rule  for 
multiplying  by  two  or  more  figures?    59.    Proof?    Principle? 


34  MULTIPLICATION. 

Kx.  20.  Multiply  5236  by  2413. 

OI'EItATIOX.  PROOf. 

Multiplicand,             5  2  3  G  2  4  13 

Multiplier,                 2413  5  2  3  6 

15  7  0  8  TT~4~7& 

5236  7239 

20944  4826 
10472                               12065 


Product,         12634468         =r  126344GB 

22. 
12474893 
7* 


Multiplicand, 
Multiplier, 

21. 
2640873 
4622 

34678 
54 

24. 
54327 
32  1 

2,5.  26. 

8645  357  9 

4  6  3  2  1  I 


27.  Multiply  4276  by  356.  Ant,  1522256. 

28.  Multiply  5463  by  248. 

29.  Multiply  4628  by  336. 

30.  Multiply  3874  by  846. 

60.  The  sign  of  multiplication,  X>  signifies  that  the  two 
numbers  between  which  it  standi  are  to  be  multiplied  together; 
thus,  6  X  5  =  30,  i.  e.  six  multiplied  by  five  equals  thirty.;  or, 
more  familiarly,  six  times  five  are  thirty. 

31.  How  many  are  726  X  W  ?  An<.  19602. 

32.  How  many  are  4628  X  554?  Ans.  2563912. 

33.  3648  X  36  =  how  many?  Ant,  131328. 

34.  4275  X  54  =  how  many?  Ans.  230850. 

35.  3759  X  8463  =?  Ans.  31812417. 

36.  53642  X  63  =  ?  41.  37642  X  57  =  ? 

37.  4020  X  524=?  42.  37942  X  386  =  ? 

38.  8726  X  463  =  ?  43.  27403  X  584  =  ? 

39.  7692  X  356=?  44.  36008  X  412  =  ? 

40.  2146  X  179  =  ?  45.  81650  X  789  =  ? 

GO.    ijign  of  multiplication,  what  does  it  signify? 


Ml'LTIPLK 


35 


4G.  If  37  men  do  I  pSeM  <>f  work  in  20  days,  in  how  many 

..  ill  1  man  do  the  MUM  work  ? 
•17.    Whal  Kl  the  vain.-  of  .'-7  land,  at  $43  per  acre? 

48.  If  a  bone  CBO   travel   41    miles   per  day,  how  lar  can  ho 
1  in   17  d;i 

49.  I  low  many  yard*  of  cloth  in  29  pieces,  if  each  piece  con- 
81  yard- ? 

61.    To  multiply  by  a  composite  number. 

A  Compositi:  NUMBEH  El  the    product  of  two  or  more  num- 

.  thai  l"»  m  a  composite  number, whose  factors  are  3  and  5; 

and  12  is  a  composite  number,  whose  factors  are  2  and  G,  or  3 
and  4,  or  2.  2.  and  3. 

It  will  1).«  observed  that  a  composite  number  may  have  several 
sets  of  (acton* 

50.  If  35  men  have  $37  each,  how  many  dollars  have  they  all  ? 

The     85     men     may    be 


OPERATION. 

35  =  5  X  7. 
Multiplicand,  S  S  7 

Lai  Factor  of  Multiplier,       5 

2.1  Factor  of  Multiplier,         7 
Product,  $  1  2  9  5 


51.  Multiply  3G7  by  1G8. 

FIRST    OPERATION. 

168  =  8  X  7  X  3. 
Multiplicand, 

:  Factor  of  Multiplier, 

nd  Factor  of  Multiplier, 


separated  into  7  groups  of 
5  men  each.  Now  1  group 
of  5  men  will  have  5  times 
$37  =  $185,  and  if  1  group 
baa  $185,  evidently  7  groups 
will  have  7  times  $185  = 
$1295,  Ani.  ;  i.  e.  7  times  5 
times  a  number  are  35  times 
that  number. 

Ans.  61656. 


3G7 
8 

293  6 

7 


SECOND    OPERATION. 

168  =  4  X  7  X  G. 

3G7 


Third  Factor  of  Multiplier, 
Product, 


20552 
3 

61656     = 


1468 
7 

10276 
6 

G  1  G  5  6 


01.     What   ||  a  o.nij'UMtf  uuinbor! 
ne  Mt  of  factor..' 


May  a  composite  number  bare  more  than 


3G  MLLTII'LICATION. 

Several  other  sets  of  factors  of  1 C8  may  be  used,  and  give 
the  same  product.  Every  similar  example  may  be  solved  in  like 
manner.     Hence, 

Rule.  Multiply  the  multiplicand  by  one  of  the  factors  of  the 
multiplier,  and  that  product  by  another  factor,  and  s<>  on  until  all 
the  factors  in  the  set  have  been  taken  ;  the  last  product  will  be  the 
true  product. 

52.  Multiply  743  by  42,  i.  e.  by  7  and  6.  Ans.  3120G. 

53.  Multiply  3467  by  56. 

54.  839  X  54  =  how  many  ?  Ans.  45306. 

55.  7869  X72  =  ? 

56.  469876X81=? 

57.  478969  X  1728  =  ?  Ans.  827658432. 

58.  5387469  X  96  =  ? 

59.  987462  X  49=? 

62.  To  multiply  by  10,  100,  1000,  or  1  with  any 
number  of  ciphers  annox**!: 

1 J  r  i.  E.  Annex  as  many  ciphers  to  the  multiplicand  as  there  are 
ciphers  in  the  multiplier,  and  the  number  so  formed  will  be  the 
product. 

i ::.  The  reason  of  the  rule  is  obvious.  Annexing  a  cipher  removes 
each  ii-ure  in  the  multiplicand  one  place  toward  the  left,  and  thus  its  value 
is  made  ten  fold  (Art.  15). 

60.  Multiply  74  by  10.  Ans.  740. 

61.  Multiply  869  by  10000.  Ans.  8690000. 

62.  Multiply  4698  by  1000. 

76984  X  100000  =  ?  Ans.  7698400000. 

64.  59874  X  1000000000  =  ? 

63.  To  multiply  by  20,  50,  500,  25000,  or  any  simi- 
1  ir  number: 

Rule.  Multiply  by  the  significant  fgures,  and  to  the  product 
annex  as  many  ciphers  as  there  are  ciphers  at  the  right  of  the 
significant  fgures  of  the  multiplier. 

61.    Rule  for  multiplying  by  a  composite  number?    62.    now  is  a  number  mul- 
tipledbylO?    By  100?    Why  ?    63.    How  is  a  number  multiplied  by  20?    Why? 


Ml  LTIl'UCATIoN.  87 

Si  Multiply  NM  l.y  30.  Ans.  22G80. 

1 1  iov.  TUl  ifl  upon  the  principle  of  Art.  Gl.    The 

7  5  G  factors  of  30  are  3  and  10.     Having  multi- 

3  0         plied  by  3,  the  product  is  multipled  by  10 

22  680         h>'  annexinS  °  (Art-  c-)* 
66.  Multiply  743  by  3500. 

OPERATION. 

7  4  3 

7  The  factors  of  3500  are  7,  5,  and 

— —  100,  .-.  multiply  first  by  7,  then  by  5, 

inn         then  annex  two  Ctpb IT  . 


Product,  2  6  0  0  5  0  0 

67.  Multiply  84G93  by  480000.  Alts.  40652640000. 

68.  87G9432  X  7200000  =  ? 

•     69.  94684235  X  49000000  =  ? 

<»-l.  To  multiply  when  there  are  ciphers  at  the  right 
of  both  multiplicand  and  multiplier: 

RULE.  Multiply  the  significant  figures  of  the  multiplicand 
by  those  of  the  multiplier,  and  then  annex  as  many  ciphers  to  the 
product  as  there  are  ciphers  at  the  right  of  both  factors. 

70.  Multiply  8000  by  900. 

operation.  The   factors   of    8000   are   8   and 

8  0  00  1000,  and   those  of   900  are   9   and 

9  0  0  100.    Now.  as  it  is  immaterial  in  what 

a„.    -Vn'nTwTA  order  the  factors  arc  taken  (Art.  59, 

Ans.   <  1  0  0  0  0  0  XT      .  i  •   i     o  i     n  .i  l 

Note),  first  multiply  8  by  .'.  then  mul- 
tiply this  product  by  1000  (Art.  G2),  and  this"  product  by  100. 

71.  Multiply  730000  by  2900. 

OPERATION. 
730000 

2900 

G5  7 
146 


Product,  2117000000 


64.    Kule  when  there  are  ciphers  at  the  right  of  both  factors?    The  reason? 

4 


38  MULTIPLICATION. 

72.  Multiply  840  by  2700000.  Ans.  2268000000. 

73.  7G93000  X  569000  =  ? 

6«5.  To  multiply  when  there  are  ciphers  between  the 
significant  figures  of  the  multiplier: 

Rule.  Multiply  only  by  the  significant  figures  of  the  mul- 
tiplier, talcing  care  to  set  the  first  figure  of  each  partial  prod- 
tict  directly  under  the  figure  of  the  multiplier  which  gives  that 
product. 

74.  Multiply  5728  by  2004. 

Thi*    is    only    carrying   out    the 

operation.  principle    (in    addition)    of    setting 

5  7  23  units  under  units,  ten-    under  tens, 

200  4         etc     The    2    of   the    multiplier   is 

2~2lT92         2000' Jind  2000  timea  ;)  tan  G000> 

IIAaq  •"•    tM<*    f    of    the    partial     product 

should  be  written  in  the  thousands' 

Product,    11468892         v\.u.,.m  j.  ...  directly  under  the  2  of 

the  multiplier. 

75.  Multiply  3724  by  4008.  Ans.  14925792. 

76.  698427  X  420006=? 

77.  7800076900  X  2008040000  =  ? 

66.    To  multiply  by  9,  99,  or  any  number  of  9's: 

IU-i.r.  "S  many  0'*  to  the  multiplicand  as  there  are 

0's  in  the  multiplier,  and  from  the  number  so  formed  subtract  the 
multiplicand ;  the  remainder  will  be  the  product  sought. 

78.  Multiply  234  by  99. 

OPERATION. 

2  3  4  0  0  =  100  times  the  multiplicand. 
2  3  4=      1  time  the  multiplicand. 

2  3  16  6=    99  times  the  multiplicand,  Ana. 

79.  Multiply  3746  by  999.  Ans.  3742254. 

80.  Multiply  427  by  9999. 

C3.  Kule  for  multiplying  when  there  are  cipher*  between  the  significant  fig- 
ures of  the  multiplier?  The  reason?  66.  To  multiply  by  9?  By  99?  Rule? 
Reason? 


MLi/nri..  39 

157.    To  multiply  by  18,  14,  15,  1G,  17,  ct 

l;    ;  Multiply  by  the  right-hand  figure  of  the  multiplier, 

get  the  produrt  und.-r  (he  multiplicand,  i  I  i  ikther  TO 

add. 

81.  Multiply  «M  by  17. 

operation.  The  1989  fa  7  times  42G,  and  the  420, 

4  2  6  standing  one  place  further  to  the  left,  is  10 

2  0  -  timet  486  (Art  15),  .%  their  sum  is  17 

7 848, Am       **"  ;~ 

82.  Multiply  849  by  is.     By  l  1.    By  1G. 

In  a  similar  manner  multiply  by  102,  1005,  10009,  etc. 

83.  Multiply  24G3  by  102. 

OPERATION. 

2  4  G3        =  100  times  24G3. 
492C  = 2     " 

25  122  0  =  102     "         "    Ans. 

84.  Multiply  3248  by  104.     By  1004.     By  1008. 

68.    To  multiply  by  21,  31,  etc.  : 

Rule.  Multiply  by  the  left-hand  figure  of  the  multiplier,  set 
the  product  under  the  multiplicand,  one  place  further  to 
the  left,  and  add. 

85.  Multiply  324  by  21. 

SHORT    METHOD.  COMMON    METHOD. 

324  324 

648  21 

6  8  0  4,  Ans.  324 

648 
6  8  0  4,  Ans. 

86.  Multiply  34264  by  81.     By  41.     By  61. 
Iq  like  manner  multiply  by  201,  301,  6001,  etc. 

87.  Multiply  4237  by  501.  Ans.  2122737. 

88.  Multiply  342G5  by  801.     By  4001.     By  30001. 

C7.   To  multiply  by  13?    By  15?    By  102?    By  1005?    Reason?    GS.   To  mul- 
Uply  by  21?    By  31?    By  601?    Reason?    Why  better  than  the  common  nwtliodf 


40  MISCELLANEOUS   EXAMPLES. 


Miscellaneous  Examples  in  Multiplication. 

1.  What  cost  11  pounds  of  beef  at  9  cents  per  pound  ? 

Ans.  99  cents. 

2.  What  cost  98  tons  of  hay  at  $15  per  ton  ?     Ans.  $1470. 

3.  In  one  hogshead  of  wine  are  G3  gallons ;  how  many  gal- 
lons in  75  hogsheads  ? 

4.  In  a  certain  house  arc  75  rooms,  in  each  room  four  win- 
dows, and  in  each  window  12  panes  of  glass;  how  many  panes 
of  glass  in  the  hou 

5.  The  eftfth,  in  its  annual  revolution,  moves  19  miles  in  a 
second;  how  far  will  it  m<»ve  in  an  hour,  there  heing  CO  seconds 
in  a  minute,  and  0<>  minutes  in  an  hour? 

G.  Light  nfoffl  I92CO0  miles  in  a  second;  how  far  will  it 
move  in  an  hour? 

7.  I  low  many  yards  of  cloth  En  10  bales,  each  bale  containing 

25  pieces,  and  each  piece  2  1  yard-? 

8.  If  12  men  do  a  piece  of  work  in  7  days,  in  how  many  d 
can  1  man  do  5  times  as  much  work  ? 

'.'.   Multiply   forty-three  million,  seven  hundred  and  four  thou- 
,!  hundred  and  sixteen,  by  forty-two  thousand  and  eight. 

10.  A  man  booghl  21  city  lots  at  $3G5  each;  what  did  they 
all  GOSl  him  ? 

11.  Multiplicand  =  4632  j  multiplier  =  4008  ;  product  =  ? 

12.  Multiplier  =  3333;  multiplicand  =  4444  ;  product  =  ? 

Examples  n  the  Foregoing  Principles. 

1.  Two  men  start  from  the  same  place,  and  travel  in  the  same 
direction,  one  at  the  rate  of  5€  miles  and  the  other  75  miles  per 
day,  how  far  apart  are  they  at  the  end  of  43  day-.? 

2.  Had  the  men  named  in  Ex.  1  traveled  in  opposite  direc- 
tions, how  far  apart  would  they  have  been  in  5G  da 

3.  Bought  o$  tons  of  hay  for  $G00  and  sold  it  for  $12  per 
ton  ;  did  I  gain  or  lose  ?     How  much  ? 

4.  Bought  25  horses  for  $125  each,  and  14  pairs  of  oxeix  at 
a  pair  ;   what  did  I  pay  for  all  ? 


■LLAHBOin   EXAMPLES.  41 

Bought  5G  barrels  of  flour  at  :-rcl.  and  in  pay  for 

COrdfl  of  r  cord,  and  the  rati  in  UK 

how  mnefa  money  did  I  | 

SCO   for  I  and  sold  the  flock  for  $425  ; 

did  I  gain  or  lose?     How  much? 

7.  A  tanner  sold  56  boahels  of  wheal  at  $2  per  bushel,  for 
which  he  received  40  yards  of  cloth  at   $2  per  yard,  and  the 

balance  in  money  ;   how  much  money  did  he  n  OB1T1  ''■ 

8.  A  merchant  bought  84G  barrels  of  Hour  for  $7191  ;  he 
sold  526  barretl  r   barrel,  and   the   remainder  at  $8  per 

barrel ;  did  be  gain  or  lose  \    Sow  much?    Ana.  Gained  S103. 

9.  A   man's   iocOJDM   i-  (1572  a  year,  and  his  expen-es  a; 

a  day;  what  do  in  a  year  of  865  days?      Ans.  $180. 

10.  Bought  18  tons  of  iron  at  $39  a  ton,  and  27  tons  at  $41  ; 
what  >hall  I  Lrain  by  Belling  the  whole  at  $13  a  ton? 

11.  A  drover  bought  a  herd  of  33  oxen,  pajing  as  many  dol- 
lars for  each  ox  as  there  were  oxen  in  the  herd.  He  paid  $500 
in  money,  ami  gave  his  note  for  the  balance  ;  what  was  the  size 
of  the  note  ? 

1  •_>.   How  many  are  8  +  2  X  7  —  3  X  5  ?  Ans.  7. 

13.  How  many  are  9X7+3x5-12?  Ans.  G6. 

1  1.   How  many  are  48  —  3  X  6  —  4  ?  Ans.  2G. 

1  ■">.  The  factors  of  one  number  are  20,  14,  and  23,  and  of 
another  1C,  8,  and  7;  what  is  the  difference  of  the  two  num- 

Ans.  5544. 
1G.  The  President  of  the  United  States  receives  a  salary  of 
1  >  a  year  ;  what  will  he  save  in  a  year  of  3 Go  days,  if  his 
expense!  are  S.")()  a  day  ? 

17.  A  man  having  a  journey  of  313  miles  to  perform  in  6 
day-;,  travels  .">  1  miles  a  day  for  0  days  :    how  far  must  he  go  on 

18.  A  many  sold  three  farms  -  for  the  first  he  reeeiv- 

for  the  second,  than  for  the  first,  and  for  the  third,  he 

ta  much  as  lor  the  other  two;  how  much  did  he 

r  the  three  fail: 

19.  What  shall  1  pay  for  25  horses,  at  $75  each,  and  12  oxen, 
at  |54  each? 

4* 


42  DIVISION. 

20.  If  a  teacher  receives  a  salary  of  $800  a  year,  and  pays 
$210  a  year  for  board,  $75  for  clothing,  $o0  for  books,  and  $100 
for  other  expenses,  how  much  will  he  save  in  3  years  ? 


DIVISION. 

60.  Division  is  the  process  of  finding  how  many  time3  one 
number  is  contained  in  another. 

The  Dividend  is  the  number  to  be  divided. 

The  Divisor  is  the  number  by  which  to  divide. 

Tin-  QtJOTll  m  \$  the  number  of  times  the  dividend  contains 
the  divisor. 

If  the  dividend  does  not  contain  the  divisor  an  exact  number 
of  times,  the  part  of  the  dividend  whieli  is  left  is  called  the 
Ki:mai.\[)!i;. 

Notk.  The  remainder  is  always  of  the  same  kind  as  the  dicidend,  because 
it  is  a  part  of  the  dividend. 

Ex.  1.  How  many  oranges,  at  4  cents  each,  can  be  bought  for 
12  cents? 

An>.  As  many  oranges  as  there  are  times  4  cents  in  12  cents; 
•1  eentfl  are  contained  in  12  cents,  3  times;  .*.  3  oranges,  at  4 
cents  each,  can  be  bought  for  12  cents. 

2.  How  many  apples,  at  2  cents  each,  can  be  bought  for  10 
cents  ? 

A  D&  A<  many  as  there  are  times  2  cents  in  10  cents,  or  as 
there  are  times  2  in  10,  viz.  5. 

70.  The  sign  of  division,  ~-,  indicates  that  the  number  be- 
fore it  is  to  be  divided  by  the  number  after  it ;  thus,  8  -f-  2  =  4, 
i.  e.  8  divided  by  2  equals  4,  or  2  in  8,  4  times. 

8.    How  many  are  6  -v-  2  ?  Ans.  2  in  6,  3  times. 

G9.  What  is  Division?  What  the  Dividend?  Divisor?  Quotient?  Remain- 
der? Of  what  kind  is  the  remainder?  70.  The  sign  of  Division,  what  does  it 
indicate? 


DIVI 


43 


In  the  same  manner,  let  the  pupil  ftTphrifl  and  recite  the  fol- 
lowing 

DIVISION  TABI 


1  -5-  1  =1 

2-4-2=1 

3-4-3  =  1 

4-4-4  =  1 

2  - 

h  1  =2 

4-4-2  =  2 

6-4-3  =  2 

8-4-4  =  2 

3  - 

h  1  =3 

64-2  =  3 

9  -4-3  =  3 

12  -4-  4  =  3 

4- 

-1  =  4 

8-4-2  =  4 

12  -h  3  =  4 

1G-4-  4  =  4 

-  1  =5 

10  ~  2  =  5 

15  -4-3  =  5 

20  -4-  4  =  5 

6- 

-1=6 

12-4-2  =  6 

18  -4-  3  =  6 

2  1  --  4  =  6 

7  - 

-1=7 

14-4-2  =  7 

21  -4-3  =  7 

2s  -s-  4  =  7 

8  - 

-  1  =8 

16  -v-  2  =  8 

2  1  :-  3  =  8 

h  4  =  8 

9  - 

-1=9 

18-4-2  =  9 

27  -4-  3  =  9 

36  -4-  4  =  9 

5-^-5  =  1 

6-4-6  =  1 

7-4-7  =  1 

8-4-  8=  1 

10  -4-  5  =  2 

12-4-6  =  2 

U-*- 7  =  2 

16  -4-  8  =  2 

15  -4-5  =  3 

18  -j-  6  =  3 

21  _f-  7  =  3 

24  -4-  8  =  3 

20  -4-  5  =  4 

24  -j-  6  =  4 

28  -4-  7  =  4 

32  -4-  8  =  4 

25  -4-  5  =  5 

30  -4-  6  =  5 

35  -4-  7  =  5 

40  -4-  8  =  5 

30  -4-  5  =  6 

36-4-6  =  6 

42  -4-  7  =  6 

48  -4-  8  =  6 

35  -4-  5  =  7 

42  -4-  6  =  7 

49  -4-  7  =  7 

06  --8  =  7 

40-4-5  =  8 

48  +  c  =  8 

56  -4-  7  =  8 

64-^8  =  8 

AS  -:-  5  =  9 

54  -4-  6  =  9 

63  -f-  7  =  9 

72  +8—9- 

9  -r-  9  =  1 

10-4-10=1 

11  -4-11  =  1 

12-4-12  =  1 

18-4-9  =  2 

20 -j-  10  =  2 

22-=-  11  =  2 

24- 

-12  =  2 

27  -4-  9  =  3 

:  10  =  3 

33-4-  11  =  3 

36- 

-12  =  3 

36  -4-  9  =  4 

40-4-10  =  4 

44-4-11  =4 

-12  =  4 

45  -4-  9  =  5 

50-4-10  =  5 

55-4-11  =  5 

60- 

-12  =  5 

:-  9  =  6 

00-4-10  =  6 

66  -4-  1 1  =  6 

72- 

-12  =  6 

63  -4-  9  =  7 

70 -h  10  =  7 

77  -4-  11  =7 

84- 

-12  =  7 

:-  9  =  8 

80-4-10  =  8 

88-4-11=8 

96- 

12  =  8 

81-4-9  =  9 

90  -r-  10  =  9 

99-4-  11  =  9 

108- 

-12  =  9 

\.  32  are  how  many  times  4?    8  ?    2?    16? 

-  are  how  many  times  4?    6?    12?    8?    3?    16? 
G.  36  are  how  many  times  12?    6?    9?    3?    4?    2? 
7.  40  are  how  many  times  8  ?   4  ?   2  ?    10  ?    5  ?    20  ? 


44 


DIYlSiU.W 


71.    Division  is  also  indicated  by  the  co/on  ;  thus,  8  :  2  =  4. 

Also  by  writing  the  divisor  before  the  dividend,  with  a  curved 
line  between  ;  thus,  2)846,  or  thus,  2  )  8  4  C  (  ,  the  quotient  to 
be  placed  under  or  at  the  right  of  the  dividend,  and  separated 
from  it  by  a  line. 

Also  by  writing  the  divisor  under  the  dividend,  with  a  line 
between  ;  thus,  $  =  3;  i.  e.  6  divided  by  2  equals  3  ;  or,  more 
familiarly,  2  in  G,  3  times. 

Ex.  8.  IIow  many  are  §?  Ans.  2  in  8,  4  times. 

The  fourth  mode  of  indicating  division  gives  the  the  following 
compact  and  convenient 

DIVISION  TABLE. 


1 

|  =  1 

1  =  1 

*  =  1 

*  =  1 

2 

*  =  2 

$  =  2 

1=2 

¥  =  2 

3 

5  =  3 

=  3 

V  =  3 

^3 

4 

|  =  4 

=  4 

V  =  4 

Y  =  4 

5 

If  =  5 

¥  =  5 

V-  =  5 

V  =  5 

G 

^G 

^G 

V  =  c 

V  =  c 

7 

V  =  7 

y.  =  7 

V  =  7 

=  7 

8 

=  8 

-^8 

-8 

¥  =  8 

9 

Jj/i  =  9 

=  9 

^  =  9 

:^9 

V  =  2 
V=3 

V  =  4 

V  =  5 

V  =  6 

V  =  9 


f=1 

1  =  1 

8  =  1 

l::  =  i 

n  =  ) 

>,*  =  2 

=  2 

¥■  =  •2 

«  =  2 

.!i'=2 

V  =  3 

^  =  3 

V=3 

?8  =  3 

^3 

3,1  =  4 

•v  =  -* 

■V  =  4 

«  =  4 

n  =  4 

V  =  o 

^  =  5 

¥  =  5 

«  =  « 

if  =  5 

^  =  G 

V  ==  6 

V  =  « 

H  =  6 

It  =  d 

V-  =  " 

¥  =  7 

=  7 

;::  =  7 

il  =  < 

V  =  8 

=  8 

=  8 

=  8 

??  =  » 

"A  s=  9 

=  9 

=  9 

H  =  9 

«  =  9 

«==  8 
'5  =  3 
H  =  4 
ff  =  5 

H  =  6 

tt==7 

|}  =  8 

=  9 


71.  Second  sign  of  Division,  what  is  it?  Third  mode  of  indicating  Division,' 
what  is  it?  Where  is  the  <;uotieut  to  be  written?  Fourth  method,  whaf  How 
are  the  dividend  and  divisor  written  in  the  second  Division  Table? 


division*.  45 

Ex.  9.   How  many  6,  or  «0«  ?  Ans.  4. 

10.  How  many  arc  46  -:-  8,  or  -*gft  ? 

1 1.  How  many  are  GO  -:-  1 1,  or  f  f  ? 

12.  How  many  are  84  -f-  12,  or  j 

1.:.  How  many  are  68  —  9,  or  «„  Ans.  7. 

1  1.  How  many  are   i-S  -f-  6,  or  ^  ? 

1">.  How  many  air  77  «+■  11,  or  \\? 

1  «'•.  How  many  arc  72  -f-  8,  or  ^  ? 

17.  How  many  an-  H    I     12,  or  ^  ?  Ans.  8. 

18.  How  many  are  88  -+■  8,  or  ^  ? 

19.  How  many  an-  72  -f-  12,  or  ^  ? 

72.  When  the  dividend  is  large  the  division  may  be  per- 
formed in  two  ways,  as  follow-  : 

2a  Divide  1384  by  4. 

nnsT  operation.  Having  written  the  divisor  and  divi- 

4)1384(346         dend  as  in  the  margin,  we  first  inquire 

1  2  how  many  times  4   is  eontained  in  13, 

7~7>  (the   fewest   figures   at  the   left   of  the 

dividend  that  will  contain  the  divisor,) 

and  find  the   quotient   to   be  3,  which 

2  4  tve   set   at    the  right    of    the   dividend. 

2_4  V,\-   then   multiply  the   divisor  by  the 

q  quotient,  3,  and    set    the    product,   12, 

under  the  13  of  the  dividend,  and  sub- 
tract it  therefrom.  To  the  remainder,  1,  we  annex  8,  the  next 
figure  of  the  dividend,  find  then  inquire  how  many  times  the 
divisor  is  contained  in  18,  the  second  partial  dividend;  there- 
suit,  1.  we  set  as  the  second  figure  of  the  quotient,  and  then 
multiply,  subtract,  annex,  etc.,  as  before,  until  all  the  figures  of 
the  dividend  have  been  taken. 

the  13  of  the  dividend  ia  hundreds,  the  3  of  the  quo- 
tient is  also  hundreds  ;  since  the  18  is  tens,  the  4  is  also  tens  ; 
and,  iinircrsalhj,  any  quotient  figure  is  of  the  same  order  as  the 
right-hand  figure  of  the  dividend  taken  to  obtain  that  quotient 
figure 

73.  How  many  ways  to  perform  Division*     Of  what  order  is  any  quotient 


46  DIVISION. 

The  foregoing  operation  is  called  Long  Division,  but  the  work 
may  be  much  shortened  by  carrying  the  process  in  the  mind,  in- 
stead of  writing  it ;  thus, 
second  orERATiox.  having  written  the  divi- 

Divisor,  4)1384  Dividend.     %    80r  and  dividend  u 
Quotient,  3  4  6  fore,  Bay,  1  in  13,  3  times 

and  1  remainder;  set  the 
quotient,  3,  under  the  3  of  the  dividend,  and  then,  innu/i/dng  the 
remainder,  1,  placed  before  the  8,  say,  4  in  18,  4  times  and  2 
remainder ;  set  down  the  4  as  the  second  figure  of  the  quotient, 
and  imagine  the  2  set  before  the  next  iigure,  and  so  proceed. 

This  operation  is  called  Short  J)lvisiun,  which  is  usually 
adopted  when  the  divisor  is  so  small  that  the  process  may  be 
readily  carried  in  the  mind.      Hence, 

73.    To  perform  Short  Division  : 

Rule.  Divide  the  left-hand  figure  or  figures  of  the  dividend, 
(the  fewest  figures  in  the  dividend  that  will  contain  the  divisor,) 
and  set  the  quotient  under  the  right-hand  figure  taken  in  the  divi- 
dend ;  if  anything  remains,  prefix  it  mentally  to  the  next  figure 
in  the  dividend,  and  divide  the  number  thus  formed  as  before,  and 
so  proceed  tUl  all  the  figures  of  the  dividend  have  been  employed. 

Ex.  21.  Divide  248G4  by  8. 

OPERATION. 

Divisor,     8)24864  Dividend. 
Quotient,     3  10  8 

22.  Divide  3240  by  2.  Ans.  1623. 

23.  Divide  1326  by  3.  Ans.  442. 
1 1.  Divide  72345  by  5.  Ans.  14 169. 
2&   Divide  3283  by  7.  Ans.  469. 

26.  Divide  59684  by  4.  Ans.  14921. 

27.  Divide  69545  by  5.  Ans.  13909. 

28.  Divide  36945  by  9.  Ans.  4105. 

29.  Divide  27512  by  8.  Ans.  3439. 

7».   What  is  the  first  method  of  Division  called?    What  the  Second?    When  i» 
Short  Division  employed?    73.  Rule  for  Short  Division? 


division.  47 

30. 

Divisor,  8)7641  28     Dividend  j  1  8  GJ_2 

QBPtfenlj  9  5516  7  1214 

32.  33.  -I. 

G ) 3  2  4  9  6         2 ) 1 48G50 B  I  4  5  8  2  8  9  2  7 

74.  When  then'  i>  no  remainder,  as  in  the  first  thirty-four 
example,  the  division  is  complete.  The  dividend  is  then  said 
divisible  by  the  divisor,  and  the  divisor  is  called  an  exact 
dir>> 

When  there  is  a  remaindi  t\  as  in  Ex.  .35,  the  di\  -i.-ion  is  in- 
complete, and  the  dividend  is  said  to  be  indivisible  by  the  divisor. 

35.  Divide  2781  by  8. 

OPKUATION. 

Divisor,     8)2781     Dividend. 

Quotient,     3  4  7  ...  5  Remainder. 

36.  Divide  3654  by  4. 
Divide  72584  by  5. 

38.  86471  -f-  3  =  how  many  ? 

39.  40505  ~  7  =  ? 

40.  476589-^9=? 

41.  987654  -J-  12  =  ? 
1l>.  334523-^-11=:? 

15.  In  one  week  there  are  7  days;  how  many  weeks  in  255 

Ad-.  36  weeks,  Rem.  3  days. 

1  L  llnw  many  barrels  of  flour,  at  $0  a  barrel,  can  be  bought 
for  S7:-<»? 

■15.  It'  6  shillings  make  a  dollar,  how  many  dollars  are  there 
In  27.3G  -hillmgs? 

46.  It  1  weeks  make  a  month,  how  many  months  are  there  in 
69  i  \\. 

74.  Wbtl  i-  t lie  division  complete?  When  is  one  number  divisible  by  an- 
other!    What  is  an  exact  divi«or!    When  ia  one  number  %ndivi$ible  by  another! 


>aoticnts, 

Rem. 

913, 

2. 

14516, 

4. 

28823, 

2. 

5786, 

3. 

48  mvr?iox. 

75.  When  the  divisor  is  large,  the  operation  is  usually 
performed  by  Long  Division,  as  follows : 

Ex.  47.  Divide  2875  by  23. 

operation.  This  operation   is  like  the  first 

2  3  )  2  8  7  5  (  125  operation  in  Ex.  20.     The  partial 

2  3  dividends  are  28,  57,  and  115  ;  the 

.-  j  successive  quotient  figures  an-  I,  2, 

a  g  and  5,  and  these  quotient  fig 

-  multiplied  into  the  divisor,  give  23, 

11^  46,  and  115  for  the  successive  prod- 

*  1  J  ucts  or   subtrahends  and    the    last 

0  product,    115,  taken  from  the   last 

dividend,  1 15,  leaves  no  remainder; 

.-.  125  is  the  true  quotient.  Ihnce, 

76.  To  perform  Long  Division  : 

Kri.i:.  1.  Write  the  divisor  and  dividend  as  in  sliort  division, 
anil  draw  a  curved  line  at  the  fight  of  the  dividend. 

2.  Divide  tl  nwmb  r  of  figvru  in  the  left  of  the  divi- 
dot'!  thai  vill  tht  divisor,  and  set  the  result  as  the  first 
ft jure  of  the  quotient  at  the  right  of  the  dividend. 

3.  Multiply  the  divisor  by  the  quotient  figure,  and  set  the  product 
under  that  part  of  the  dividend  t<iken. 

4.  Subtract  the  product  from  the  figures  over  it,  and  to'the  re- 
mainder annex  the  next  figure  of  the  dividend  for  a  new  partial 
dividend, 

5.  Divide,  and  proceed  as  before,  until  the  whole  dividend  has 
been  divided. 

Note  1.  It  will  be  seen  that  the  process  of  dividing  consists  of  four  dis- 
tinct steps,  viz.:  fait,  to  seek  B  quotient  figure;  second,  multiply;  third, 
inhered  ;  and,  fourth,  form  a  new  partial  dividend  by  annexing  the  next 
figure  of  the  dividend  to  the  remainder. 

NOT!  2.  If  any  partial  dividend  will  not  contain  the  divisor,  0  must  be 
pieced  in  the  quotient,  and  another  figure  annexed  to  the  partial  dividend. 

Note  3.     If  the  product  of  the  divisor  multiplied  by  the  quotient  figure 

7.-,  When  k  Long  Division  employed?  Explain  Ex.  47.  70.  Give  the  rule 
for  Long  Division.  How  many  steps  in  dividing?  What  arc  they?  Repeat 
Note  2.    Not*  3.    Not*  4. 


J»IV!  (ft 

Ttor  than  the  partial  dividend,  the  quotient  figure  is  too  large,  and 
tbo  diminished. 

i.  4.     If  the  remainder  equals  or  exceeds  the  divisor,  the  quotient  is 

i;ill,  nnd  must  1><- 

77.  Division  is  (he  reverse  of  multiplication.  In  multiplica- 
tion the  two  &Ctor  li.  ami  the  product  is  required  j  in 
division  the  product  and  one  factor  are  given,  and  the  other  f;l.-- 

-  required.     The  dividend  ia  the  product,  and  the  diruor 

and  quotient  are  tlu*  factors  ;  thus, 

IN    Mil  1  Ii'LICATIOX.  IN    IMVISIOIf. 

Factors,       Product.  Dividend,         Divisor,         Quotient 

5x  4  =  10  20      -f-      5       =       4 

Or,  20      -J-      4       =       5 

Hence  the 

78.  Proof.  Multiply  the  divisor  by  the  quotient,  and  to  the 
product  add  the  remainder  ;  the  sum  should  be  the  dividend. 

48.     Divide  2537  by  53. 


OPERATIOX. 

53)2537(47 
2  1  2 

417 
371 

46 

PROOF. 

5  3  Divisor. 
4  7  Quotient. 

371 

212 

4  6  Remainder. 

25  3  7  Dividend. 

49. 
21  )864(41 
84 

24 
21 

50. 
87)3659(42 
348 

179 
174 

3 

5 

51.  A  flock  of  1728  sheep  were  divided  equally  in  9  different 
lAttarea,  how  many  ibeep  were  there  in  each  pasture? 

77.    What  is  said  of  Division  and  Multiplication?     In  Multiplication  what  is 
'     What  required ?     In  Division  what  is  given?    Required?    78.    How  is 
:i  proved? 

5 


Quotients. 

Rem. 

1509, 

j6. 

2615, 

16. 

10175, 

24. 

5075, 

43. 

10475, 

7. 

ii 

11237, 

57. 

1090, 

124. 

50  DIVISION. 

.  Divide  46782  by  31. 

53.  Divide  47086  by  18. 

54   Divide  468074  by  46. 

1 10068  by  67. 

!  -h  83  =  how  manv  ? 

57.  9  ;7048-^99=rhowmauy? 

+  78  =  BOW  many? 

59.  276984  -*-  254  =  ? 

CO.  376958-4-84$=? 

81.  876598-4-427=? 

61  469873^789=? 

9  -s-803  =  ? 

64  896842  ~  548  =  ? 

,:*2^45=r? 

\  ~  4893  =  ? 

5=? 

68.  Divide  four  hundred  eighteen  thousand,  six  hundred  and 
•€ight,  by  tw.nty-four.  An?.,  Quo.  17443,  Rem.  16. 

69.  Divide    two    hundred    one    thousand,    live    hundred   and 
ninety-live  acres  of  land,  into  twenty-three  equal  parts. 

7<».  A   railroad  that  cost  f  divided  into  7153 

equal  shares  ;   what  was  the  cost  of  etch  share? 

71.  A  farmer  raised  2001  bushels  of  wheat  on  87  acres  of 
land ;  how  many  bushels  did  he  raise  per  acre  ? 

72.  In  how  many  days  will  a  ship  sail  34">6  miles,  if  it  sails 
1  I  1  miles  per  day? 

A  farmer  raised  4088  bushels  of  corn,  his  crop  averaging 
56  bushels  per  acre;  how  many  acres  did  he  plant  ? 

7  1.  A  drover  paid  $3175  for   29  oxen;    how  many  dollars 
did  he  pay  for  each  ox  ? 

7 "'.    The  product  of  two  numbers  is   10707,  and  one  of  the 
numbers  is  129  ;  what  is  the  other  number? 

The  earth,  in  its  revolution  round  the  sun,  moves  about 
1641600  miles  in  one  day  ;  how  far  does  it  move  in  one  second, 
there  being  86400  seconds  in  a  daj  \ 

77.   Divide  §1064  equally  among  8  men.  Ans.  S133. 


PI  VI 


51 


79.   To  divide  by  a  composite  number. 
.  78.  Divid  equally  among  85  men. 

on 

z  7  X  5.  The  85  men  may 

ictor,  7  )  $  1  8  5  5  Dividend.  be  separated  into  1 

M  Emctor,      5)S265  1st  Quotient,  -r",u'"    'f    *   .*" 

' ^  each,      riicn  divid- 

$5  3  True  Quotient.        fog  by  7  giv< 

for  each  group,  and 
dividing  the  $2G.">  by  5  gives  S">3-  for  cacli  man. 

Vhcn  a  eomposito  numl>er  is  made  up  of  different  sets  of  far- 
tors,  os  in  Ex.  79,  it  is  immaterial  which  sot  is  taken.  It  is  also  immaterial 
in  what  order  the  factors  arc  tuken. 

79.  Divide  10G5G  by  288. 
288  =  4  X  6  X  12  =  6  X  6  X  8  =  8  X  3  X  12,  etc. 

FIRST    OPKRATIO.V. 

4)  10656 


BECOJCD    OPERATION. 

G  )  1  0  6  5  6 


664 

1  2  )444 
37 


C)1776 

8)296 

"77 


From  these  examples  we  have  the  following 

Kile.     Divide  ike  dividend  by  one  factor  of  the  divisor,  and 

the  quotient  so  obtained  by  another  factor,  and  so  on  till  all  the 

the  set  have  been  used.      The  last  quotient  will  be  the 

true  quotient. 


80.  Divide  1551  by 

81.  Divide  31794  by  42. 
Divide  47986  by  5G. 

88.  Divide  24840  I 

M.  Divid."  7665  by  L05, 

Divide  1064  by  5G. 

Divide  1984  by  64 
87.  Divide  8321  by  81. 


88.  Divide  18723G  by  252. 

89.  Divide  1255872  by  192. 
i»o.  Divide  1865  by  105. 

91.  Divide  :>:5.V>  by  315. 

92.  Divide  GG99  by  281. 

93.  Divide  8822  by  294. 

94.  Divide  s:>i}8by504. 

95.  Divide  7245  by  315. 


7'J.  Bali  for  dividing  by  a  Composite  Number?    la  it  material  which  factor  of" 
the  divisor  is  used  flrkt? 


52  DIVISION. 

80.  In  dividing  by  the  factors  of  the  divisor,  there  may  be  a 
remainder  after  either  or  each  of  tin*  divisions. 

Should  the  learner  find  a  difficulty  in  determining  the  true  re- 
mainder, he  has  but  to  remember  that  it  is  always  of  the  same 
kind  as  the  dividend  (Art.  G9,  Note). 

96.  Divide  86  by  lh 

OPERATION. 

7  )  8_6  In  this  example,  as  86  is  the 

true    dividend,    2    is    the  true 
remainder. 


In  this  example,  as  23  is  only 
one  fourth  of  the  true  dividend, 
so  the  remainder,  2,  is  only  one 
fourth  of  the  true  remainder; 
.•.2X4=8,  true  remainder. 


By  the  explanation  of  exara- 
7  Tg~7       5  j^eni  pies  96  and  97,  we  see  that  5  is 

'  — " "  one  part  of  the  true  remainder, 

Quotient,  1  %  . . . 8  Rem.  an(j  tbmt  5,  the  second  remain- 
der, multiplied  by  6,  the  Ant 
divisor,  is  the  other  part;  i.  e.  5  -\-3  X  &  =  23,  is  the  true  re- 
mainder. The  same  species  of  reasoning  applies  when  there  are 
more  than  two  divisors.     Hence, 

To  obtain  the  true  remainder  when  division  is  per- 
formed by  using  the  factors  of  the  divisor: 

Rite.  Multiply  each  remainder,  except  that  left  by  the  first 
division,  by  the  continued  product  of  the  divisors  preceding  that 
which  gave  the  remainders  severally,  and  the  sum  of  the  products, 
together  with  the  remainder  left  by  the  first  division,  will  be  the 
true  remainder. 

Note  1.     "When  there  are  but  two  divisors  and  two  remainders,  the  rule 

80.  Rule  for  finding  the  true  remainder  when  the  factors  of  the  divisor  ar« 
used  separately?    The  reason?    What  is  meant  hy  a  continued  product? 


3)12... 

2  Rem, 

Quotient,            4 

Divide  9! 

OPERATION. 
7)23 

28. 

Quotient,            3  . . . 

2  Rem. 

98. 

Divide  5S7  bj 

OPERATION. 
6)527 

DIVISION.  53 

only  requires  the  addition  of  the  Jirst  remainder,  to  the  product  of  the  firtl 
and  second  return 
Rbl  I  2.     When  thnf  or  more  factors  are  multiplied  together,  the  product 
is  called  a  continued  product. 

Quo.  52,  Rem.  14. 

I  1:1    1.     KIM  \IM 

4= lei  Rem. 

<  5  =  10  =  2d  Rem.  X  1st  Div 
14  =  True  Rem. 


TRUE   REMAINDER. 

3  =  1st  Rem. 


99. 

Divide  1884  by  35. 

OPERAT1 

-5X7. 
5)1834                              \ 

7  )  3  6  6  ...  4,  1st  Rem. 
Quo.,~~5~2  ...  2,  2d  Bern. 

100. 

Divide  18328  by  385 

oil  i:  vriON. 

=  5X7X11. 

5)18328 
7)3  6  65  ...  3,  1st  Rem. 

11) 

5  2  3  .  .  .  4,  2d  Rem. 

Quo., 

4  7  ...  6,  3d  Rem. 

101. 

Divide  5273  by  42. 

42  =  6  X  7. 

102. 

Divide  4G987  by  504 

103. 
104. 
105. 
10& 

1<)7. 

437298-^-54  =  ? 
21C349-t-64  =  ? 
8411  -5-72=? 

,,7_j_45  =  ? 
65947 -f- 25=? 

4X5=     2  0  =  lstProd. 
6X7X5  =  210=  2d  Prod. 
2  3  3  =  True  Rem. 


Ans.  125  and  23  Rem. 

ng  the  factors  of  the  divisor. 
Am.  88  end  115  Rem. 

108.  6842 -v- 49=? 

109.  7829 -f- 35  =  ? 

110.  8748  -#-  42h± t 

111.  4629-^81  =  ? 

112.  3643  «)L  48  =  ? 

81.    To  divide  by  10,  100,  1000,  etc.  : 

Rule,      Cut  off]  by  a  point,  as  many  figures  from  the  right 
I  of  the  dividend  at  then  are  ciphers  in  the  dirisor.      The 
•  s  at  the  left  of  the  point  arc  the  quotient,  and  those  at  the 
right  are  the  remainder. 

1 13.  Divide  75G  by  10.         Ans.  75.6,  i.  e.  75  Quo.,  6  Rem. 

81.  Rule  for  dividing  by  10?    By  100? 

5* 


54  DIVISION. 

Note.  The  reason  of  the  rule  is  obvious.  By  taking  away  the  right- 
hand  figure,  each  of  the  other  figures  is  brought  one  place  nearer  to  units, 
and  its  value  is  only  one  tenth  as  great  as  before  (Art.  15),  and  .'.  the  whole 
is  divided  by  10.  For  like  reasons,  cutting  off  ttco  figures  divides  by  100 ; 
cutting  off  Utree  figures  divides  by  1000,  etc. 

114.  Divide  402763  by  10. 

115.  Divide  7G943  by  100.  Ans.  769  and  43  Rem. 
11G.  Divide  98765423  by  100000. 

!  Ans.  987  and  65423  Rem. 

117.  Divide  3078654321  by  100000000. 

8£.    To  divide  by  20,  50,  700,  or  any  similar  number : 

Rdle.  Cut  off  as  many  figures  from  the  right  of  (he  divi- 
dend as  there  are  ciphers  at  the  right  of  the  significant  figures  of 
the  divisor,  and  then  divide  the  remaining  figures  of  the  dividend 
by  the  significant  figures  of  the  divisor. 

Note  1.  This  is  on  the  principle  of  dividing  by  the  factors  of  the  divi- 
sor ;  .-.  the  true  remainder  will  be  found  by  the  rule  in  Art.  80. 

118.  Divide  74689  by  8000.  Ans.  9  and  2689  Rem. 

operation.  AW  divide  by  1000  by  cutting 

8  )  7  4.6  8  9  off   689,  which  gives  74  for  a 

Ouotient     ~~ 9        2  Rem  quotient  and  689  for  a  reiuain- 

^  '  der ;    then  divide  74  by  8,  and 

obtain  the  quotient,  9,  and  remainder,  2.     This  remainder,  2,  is 

2000,  which,  increased  by  689,  gives  2689  for  the  true  remainder 

(Art.  80). 

Note  2.  It  will  l>c  observed  that  the  true  remainder,  in  all  examples 
/ike  thollSth,  is  obtained  by  annexing  the  1st  to  the  2d  remainder. 

119.  Divide  07475  by  2400. 

120.  Divide  74689  by  4200.     Ans.  17  and  3289  Rem. 

121.  Divide  276987  by  3300. 

1 22.  769842  -^-  45000  =  ?      Ans.  17  and  4842  Rem. 
i«8.  9999999-1-33300=:? 

124.  80407080  —  40000  =  ? 

125.  987654321  -f-  90900  =  ? 

81.  Beason  of  rule  for  dividing  by  10?  83.  Rule  for  dividing  by  20?  By 
LOO?    Reason?    IIow  is  tbe  true  remainder  found? 


DIVIifi  56 


Qf    Div; 

821.    The  value  of  a  quotient  depends  upon  tlio  relative 
values  of  the  divisor  and  dividend,  and   not  upon  their 
luU  values,  us  will  be  seen  by  the  following  propo- 
rtions. 

(a)  If  the  divisor  remains  unaltered,  multiplying  the  dividend 
by  any  number  is,  in  effect,  multiplying  the  quotient  by  the  same 
number  ;   thus, 

15-f-3=     5 
_4  _4 

00-^3  =  20; 

i.  «•.  multiplying  the  dividend  by  1  multiplies  the  quotient  by  4. 

(1>)  DMding  the  dividend  by  any  number  is  dividing  the 
quotient  by  the  same  number  ;  thus, 

24-4-2  =  1  2 
3)24 

8-^  2=     4=12-v-3; 

i.  e.  dividing  the  dividend  by  3  divides  the  quotient  by  3. 

(c)  Multiplying  the  divisor  divides  the  quotient  ;  thus, 

3  0-^-2  =  1.-) 

3 

naltipljiag  the  divisor  by  >°>  divides  the  quotient  by  3. 

(d)  Dividing  the  divisor  multiplies  the  quotient  ;  thus, 

40-4-  10—     I 
5)  1_0 

4  0  -f-     2  =  20=r4X-">; 

i.  e.  dividing  the  divisor  by  o  mult  plies  the  quotient  by  ">. 

B3.   Does  the  size  of  the  quotient  depend  upon  the  absolute  size  of  divisor  and 

i  ;  .hi  what  dOM  it  clrpoiul  ?     What  is  the  first  proposition?     Second  r 
•     Fourth? 


56  DIVISION. 

(e)  It  follows,  from  (a)  and  (b),  that  the  greater  the  dividend, 
the  greater  is  the  quotient ;  and  the  less  the  dividend,  the  less 
the  quotient. 

(f)  Also,  from  (c)  and  (d),  that  the  greater  the  divisor,  the 
less  is  the  quotient  ;  and  the  less  the  divisor,  the  greater  the 
quotient. 

84.    From  the  illustrations  in  Art.   83  we  see  that 
any  change  in  the  dividend  causes  a  similar  change  in 
the  quotient,  and  that  any  change  in  the  divisor  can 
an  opposite  change  in  the  quotient.     Ilence, 

(a)  Multiplying  both  dividend  and  divisor  by  the  same  number 
does  not  affect  the  quottini  :  thus, 

12-^3  =  4 
_2       2 

2  4  -r-  G  =  4,  Quotient  unchanged. 

(b)  Dividing  both  dividend  and  divisor  by  the  same  number 
does  not  affect  the  quotient ;  thai, 

20       -4-       10  =  2 
5)20  5)  H) 

4       -j-  2  =  2,  Quotient  unchanged. 

(c)  It  follows  from  (a)  and  (b),  that  the  operations  of  multi- 
plying and  dividing  by  the  same  number  cancel  (i.  e.  destroy) 
each  other  ;  e.  g., 

If  a  number  be  multiplied  by  any  number,  and  the  product  be 
divided  by  the  multiplier,  the  quotient  will  be  the  multiplicand : 
thus, 

8X7  =  56,  and  56  -J-  7  =  8,  the  multiplicand. 

Also,  if  a  number  be  divided  by  any  number,  and  the  quotient 
be  multiplied  by  the  divisor,  the  product  will  be  the  dividend ; 
thus, 

15  -f-  3  ■=  5,  and  5  X  3  =  15,  the  dividend. 

83.  What  follows  from  (a)  and  (b)?  From  (c)  and  (d)?  84.  Any  change  in 
the  dividend,  how  does  it  affect  the  quotient?  Any  change  in  the  divisor,  how? 
First  inference?    Second?   Third?   Illustrate. 


MISCELLA  SAMPLES.  6f 

S3.   These  general  principles  may  be  more  briefly 
1  as  follows: 

1-t.  Mult iplying  the  dividend  multiplies  the  quotient  ;  and 
dividing  the  dividend  diviiles  the  quotient  (Art.  83,  a  and  b). 

2d  Multiplying  the.  divisor  divides  the  quotient  ;  and  dividing 
the  divisor  multiplies  the  quotient  (Art.  88,  0  and  <1). 

Multiplying  both  dividend  and  divisor  by  the  same  num- 
ber ;  or  dividing  both  bij  the  same  number  does  not  affect  the  quo- 
( Art.  84,  a  and  b). 

Examples  in  the  Foregoing  Principles. 

i 

1.  Hon  'Many  bu>hels  of  corn  at  SI  per  bushel  mutt  be  given 

for  6  barrels  of  flour  at  $7  pel  barrel? 

2.  How  many  bands  of  apples  at  $2  per  barrel  must  be  given 
be  8  cords  of  wood  tl  $6  per  cord? 

3.  A  speculator  bought  640  acres  of  land  at  $8  per  acre,  and 

void  the  whole  for  $3200;  how  much  did  he  gain  by  the  trans- 
lation?    Bow  much  per  aen 

4.  Bought  320  acres  of  land  for  $1760,  and  320  acres  more  at 
^7  pel  acre,  and  sold  the  whole  at  $6  per  acre;  did  I  gain  or 

How  much?  Ans.  Lost  $160. 

5.  The;  expenses  of  a  boy  at  school  for  a  year  are  $126  for 
board)  $24  for  tuition,  $15  for  books,  $35  for  clothes,  $10  for 
railroad  and  coach  fare,  and  $0  for  other  purposes  ;  what  will  be 
the  expenses  of  250  boys  at  the  same  rate  ? 

('..   If  :;  men  build  24  rods  of  wall  in  4  days,  in  how  many 
ill  5  men  build  70  rods?  Ans.  7. 

7.  The  product  of  4  factors  i<  1155;  three  of  the  factor! 
8,  5,  and  7  ;   what  is  the  fourth?  Ans.  11. 

8.  How  many  miles  per  hour  must  a  ship  sail  to  CTOS 
Atlantic,  2880  miles,  in  12  days  of  24  hours  each  ? 

(J.  The  first  of  3  numbers  is  6,  the  second  is  5  times  the 
first,  and  the  third  is  4  times  the  sum  of  the  other  two;  what  is 
the  difference  between  the  first  and  third? 

W.   A  more  brief  *t«totn«>nt  of  (hew  principle  fecoad!    Third? 


68  REDUCTION.  ^ 

10.  Sold  two  cows  at  $30  apiece,  3  tons  of  hay  at  $20  per 
ton,  50  bushels  of  corn  for  $50,  and  10  cords  of  wood  at  $7  per 
cord,  and  received  in  payment  $200  in  money,  a  plow  worth 
$15,  50  pounds  of  sugar  worth  $5,  and  the  balance  in  broadcloth 
at  $4  jet  yard  ;  how  many  yards  did  I  receive?  Ans.  5. 

11.  In  how  many  days  of  24  hours  each  will  a  ship  cross  the 
Atlantic,  2880  miles,  if  she  sails  10  miles  per  hour? 

12.  If  I  receive  $60  and  spend  $40,  per  month,  in  how  many 
years  of  12  months  each  shall  I  save  $2160  ?  Ans.  9. 

13.  "What  ii  the  value  of  27  hogsheads  of  molasses  at  j  - 
per  hothead? 

14.  What  i>  the  value  of  87  yards  of  cloth  at  .ml  ? 

15.  Bought  87  acres  of  land  at  |  and  paid  $3150 
in  cash,  and  the  balance   in  labor  at  $240  a  year;  how  many 

I  of  labor  did  it  take  ? 

16.  Bought  42  yards  of  cloth  at  15  cents  per  yard,  and  paid 
for  it  in  corn  at  90  cents  per  bushel ;  how  many  bushels  did  it 
take? 

17.  If  I  take  13729  from  the  sum  of  8762  and  14967,  divide 
the  remainder  by  50,  and  multiply  the  quotient  by  19,  what  is 
the  product  ?  Ans.  3800. 


REDUCTION 


80.    All  numbers  are  simple  or  compound. 

A  Simple  Number  consists  of  but  one  hind  or  denomina- 
tion ;  as  2,  $4,  8  books,  5  men,  6  days,  10  miles. 

A  Compound  Number  is  composed  of  two  or  more  denom- 
inations ;  as  4  days  and  7  hours ;  8  bushels,  2  pecks,  and  5 
quarts ;  5  rods,  4  feet,  and  6  inches. 

All  abstract  numbers  (Art.  2)  are  simple. 

86.    What  is  a  Simple  Number!    A  Compound  Xumbcr?    An  Abstrac    dum- 
ber, is  it  simple  or  conipom.U  I 


A  concrete  UBUllMT,  whrtlit -r  simjilo  or  compound,  is  often 
called  a  Denominate  Xm 

i:  1.     All  operations  in  the  preceding  pages  are  upon  simple  num- 
bers. 

The  several  parts  of  a  compound  number,  though  of  different 

denominations,  are  yet  of  the  same  general  nature;  thus,  2  Iraki,  •'*  days,  and 

6  hours  are   similar  quanti  ^titute  a   comjiound  number;  but  2 

md  6  quarts  arc  DIUKI  in  thliu  xailue,  and  do  not 

constitute  a  compound  number. 

S7.  Induction  is  changing  ■  number  of  one  denomination 
to  one  of  another  denomination,  without  changing  its  value. 

It  is  of  two  kinds,  viz.  Reduction  Descending  and  Jteduction 
Ascending. 

Reduction  Descending  consists  in  changing  a  number 
from  a  higher  to  a  lower  denomination. 

duction  Ascending  is  changing  a  number  from  a  lower 
to  a  higher  denomination. 

ENGLISH  MONEY. 

88.    English  Money  is  the  Currency  of  Great  Britain. 

TABLE. 

4  Farthings  (far.  or  qr.)  make         1  Penny,  marked  d. 
12  Pence  u  1  Shilling,     "       p. 

20  Shillings  "  1  Pound,        "       £ 


gs 

14 

m 

1  Shilling, 
1  Pound, 

£ 

1 
20 

•  d. 
1 
=      12 
=    8 

qr. 
s=       4 
=     48 
=   960 

80.    REDUCTION  Descending  is  performed  by  multiplica- 
tion;  thus,  to  reduce  15£  to  shillings,  we  multiply  15  by  20, 
use  there  will  be  20  times  as  many  shillings  as  pounds.     So 
to  reduce  15£  and  12s.  to  shillings,  we  multiply  15  by  20,  and 
to  the  product  add  the  12s. 

86.  A  Concrete  Number,  what  is  it  called?    87.  What  is  Reduction?    How 

many  kinds  of  Reduction?    What  are  they  called?    What  is  Reductiou  Deecend- 

Itafl     88.   What  is  English  Money?     Repeat  the  table. 

N'J.    How  is  Rcduclion  l>c.-c»-ndiiij(  performed? 


00  REDUCTION. 

In  a  similar  manner  all  mob  examples  are  reduced.     II 

90.  To  reduce  the  higher  denominatious  of  a  com- 
pound number  to  a  lower  denomination: 

Rule.  Multiply  die  highest  denomination  given  by  the  number 
it  takes  of  the  next  lower  denomination  to  make  one  of  this  higher, 
and  to  the  product  add  tfte  number  of  the  lower  denomination  ; 
multiply  this  sum  by  the  number  it  takes  of  the  next  lower  denom- 
ination to  make  one  of  this  ;  add  as  before,  and  so  proceed  till 
the  number  is  brought  to  the  denomination  required. 

Ex.  1.  Reduce  11X  17s.  9d.  3qr.  to  farthings. 


Eleven  pounds  =  220s.,  and 
the  17s.  added  make  237s.  = 
2844d.,  and  the  9d.  added  give 
a  11412qr.,  which,  in- 
creased by  the  3qr.,  give  11415 
(jr.,  the  answer. 
1  1   1  i  o  qr.,   An>. 

2.  Reduce  6£  18s.  4d.  lqr.  to  farthings.  Ans.  G641qr. 

8.  1:  £  9s.  8qr.  to  farthings.  Ans.  7155qr. 

Note.  Since  there  are  no  pence  in  the  3d  example,  there  is  nothing  to 
add  to  the  product  obtained  by  multiplying  1-v  i& 

ft.   Etedoee  27£  15s.  Gd.  2qr.  to  farthings. 
5.  Reduce  32£  8d.  3qr.  to  farthings. 

01.  Reduction  Ascending  is  performed  by  division  ; 
thus,  to  reduce  4299  farthings  to  pence,  ue  divide  the  4299  by 
4,  because  there  will  be  only  one  fourth  as  many  pence  as  far- 
things. Performing  the  division  we  obtain  1074d.  and  a  remain- 
der of  3*qr.  If  we  wish  to  reduce  the  1074d.  to  .-hillings,  we 
divide  by  12,  because  there  will  be  only  one  twelfth  as  many 
shillings  as  pence,  and  obtain  89s.  and  a  remainder  of  Gd.  Again, 

93.  Kepeat  the  rule.  Explain  the  process  in  Ex.  1.  How  are  the  2G7  shil- 
lings obtained?  How  the  2S63  pence?  The  lHlo  farthings?  01.  How  is  Re- 
duction Ascending  performed? 


£           8. 

11     17 
20 

d. 
9 

2  3  7s. 
12 

2853d. 
4 

ION.  61 

tO  pounds  by  dividing  by  20,  givi: 

Hod  •  renuunder  of  9a.    Tims  ire  Sod  that  4299qr.  are  eqnal  to 
8qr. 

Llkl  >g  ippUoi  to  all  similar  mWiplfMli      II'  I 

92.  To  reduce  •  number  of  a  lower  denomination  to 
numbers  of  higher  denominations: 

BULK.      Divide  the  given  number  by  the  number  it  tales  of  that 
denomination  to  male  one  of  the  next  higher;  divide  the  quotient 
number  it  takes  of  THAT  denomination  to  make  one  of  the 
/<i>//ter.  and  so  proceed  till  the  number  is  brought  to  the  de- 
nomination required.     The  last  quotient,  together  with  the  several 
remainders  (Art.  69,  Note),  will  be  the  uns> 

93.  Redaction  Ascending  and  Reduction  Descending  prove 

each  other. 

Ex.  1.  Reduce  11415  farthings  to  pence,  shillings,  and  pound-. 

araanox.  First  divide  by  4  to  reduce  the 

A  )  1  1  4  1  o  qr.  farthings  to  pence;  then  divide  by 

1  2  i~2  8  5  3  d  -4-3or  *^    t0    re(^uce    Pence    t0    shillings  ; 

'  then  by   20    to    reduce   shilli: 

2  0  )  2  3  7  s.  +  9d.  pounds,  and  thus  obtain   ll£   17s. 

1  l£+17s.  &  .'Mir.,  Ana, 

2.  Reduce  17229qr.  to  pence,  shillings,  and  pounds. 

.   17£  18s.  lid.  lqr. 
o.  Reduce  6874d.  to  shillings  and  pounds. 

Ana.  28£  12a,  lOd. 

I  1.  Since  Ex.  3rd  is  givm  in  pence  instead  of  farthings,  the  first 
divisor  is  12  rather  than  4. 

4.  Reduce  84697qr.  to  higher  denomination-. 

Etedooa  124G83qr.  to  higher  denominations. 
G.   Reduce  3-17G2  Iqr.  to  pence,  shillings,  and  pounds. 
7.   Reduce  8746d.  to  shillings  and  pounds. 
\  to  pounds. 

'J  1.    iapeat  the  ru!f.    Explain  the  process  in  Kx.  1.     How  are  the  8qr.  olr: 
Tli.lTr.?    TliclU?    03.   What  is  the  J'/ -  tloaJ 

8 


REDUCTION. 


Note  2.  The  numbers  employed  in  the  reduction  of  a  compound  num. 
ber  are  called  a  Scale.    The  descending  scale  for  Reduction  Descend' 

in j  and  an  ascending  scale  for  Reduction  Ascending;  thiu,  in  English  money 
the  desceudinq  scale  is  20,  12,  and  4,  and  the  ascending  scale  is  4,  12,  an<l  Mfc 
The  descending  scale  consists  of  the  numl>crs  at  the  left  hand  of  the  table, 
taken  in  order  from  the  bottom  to  the  top  of  the  tabic,  and  the  ascending  scald 
consists  of  the  same  numbers  taken  in  the  reversed  order,  i.  e.  from  the  top  to  tha 
bottom  of  the  table.     In  like  manner  the  scale  is  found  in  the  other  tables. 

TROY  WEIGHT. 

04.    Trot  Weight  is  used  in  weighing  gold,  silver,  and 
precious  stones. 


TABLE. 


24  Grains  (gr.) 
20  Pennyweights 
12  Ounces 


make 

u 


1   Pennyweight,      dwt. 
1  Ounce,  oz. 

1   Pound,  lb. 


lb. 

1 


oz. 

1      = 

12        = 


dwt. 

1  = 

20  =: 

240  = 


24 

480 

5760 


Ex.  1.  How  many  grains  in 
71b.  lloz.  14dwt.  18 

OPERATIOX. 

71b.  lloz.  14dwtl8gr. 
12 

9  5oz. 
20 


Ex.  2.    Reduce  45954gr.  to 
pounds,  ounces,  etc. 

OPERATION. 

24)4595  4gr. 
20)  1914dwt-f-18gr. 
12)9J5oz.    -j- 14  dwt. 
7  lb.    +11  oz. 


Ans.71b.  lloz.  14dwt.  18gr. 


19  14  dwt. 
24 

7674 
3828 

~>  4  gr.,    Ans. 

Note  1.  In  solving  Ex.  1,  the  several  numbers  of  the  lower  denomina 


93.  What  is  a  scale?  A  descending  scale?  An  ascending  scale?  What  ar» 
the  scales  for  English  money?  Where  are  these  scales  found?  Taken  in  what 
order?  94.  For  what  is  Troy  Weight  used?  Bepeat  the  table.  Descending  scaler 
Ascending? 


[0», 

dons  are  added  vimtallif,  and  only  the  i  rittm  ;  thus,  12  ti 

are  84,  and  the  llos.  a.  1  >/.      Then  multiplying  tin-  Moa 

I  Ling  the  1  -l 1 1  w r . .  If]  idwt.     Finally,  in  multiplying  the  19U 

dwt.  by  M,  list  multiply  by  4,  adding  in  tin-  ISgr.,  and  then   multiplying 

uitl  adding  thi  results  wt  hm\  wtr, 

•_\  if  any  divisor  is  so  large  that  the  work  is 
lvdono  by  Short  l)ivi<ion,  the  numbers  mayl>c  taken  upon  the  llnH 
and  the  work  done  by  Long  Division,  setting  down  only  the  results. 

I  I.»w  many  grains  in  101b.  8oz.  19dwt.?     Ans.  964a6gr. 
4.  Reduce  386'Jogr.  to  pounds,  etc 

Ans.  61b.  8oz.  12dwt.  7gr. 
ft.   Bedaee  87942gr.  to  pounds,  ounces,  etc 
lb.  8oz.  6dwt  16gr.  to  grains. 

7.  Bow  many  BpoOQtj  each  weighing  2oz.  8dwt.  20gr.,  can  b« 
made  from  21b.  Ooz.  6dwt  of  silver?  Ans.  12. 

8.  A  jeweller  made  8oz.  16dwt.  of  gold  into  rings  wbicb 
weighed  8dwt  16gr.  eacb  ;  bow  many  rings  did  be  make  ? 

APOTHECARIES'    WEIGHT. 

93>.  Apothecaries'  Weight  is  used  in  mixing  or  com- 
pounding medicines  ;  but  medicines  are  bought  and  sold  by 
Avoirdupois  Weight. 

TABLE. 


20  Grains  (gr.) 
3  Scruples 
8  Drams 

19  Ounces 

make          1  Scruple,     sc.  or  9 

"              1  Dram,        dr.  or  3 

"             1  Ounce,       oz.  or  5$ 

1  Pound,        lb.  or  lb 

oz. 

lb.           1 

1       =       12 

sc.                            gr. 

dr.  1  BS  20 
1  =  3  =  60 
8           =        24          =        480 

96           ss      288          =      5760 

::    1.     The  pound, 
it  are  equal,  but  the 

ounce,  and  grain,    in  Apothecaries'  and  Troy 

•  differently  subdivided. 

9*.    In  solving  Ex.  1,  what  is  done  with  the  numbers  of  the  lower  denomina- 
te 2,  how  is  the  work  done?    95.    For  what  is  Apothecaries'  W.  Iftt 
Used?     Repeat  the  tabic.     Descending  scale?    Ascending?    What  denomination:! 
•f  Apothecaries'  Weight  ars  like  thow  of  Troy  Weight? 


Bl  AUCTION. 

I.x.  1.   How  many  scruples  .2.   In  13619  how  many 

in  41b  8§  53  29  ?  pounds,  oun< 


VTIOX. 


4  lb  8§  53  29  3  )  1_3  G  1  9 

11  8)4533  +  29 

8§  It)£SB  +  5  5 

TT^3  4lb+85 
3 


13  6  19,     Aim. 


Aim.   41b  6j  53 


3.  Reduce  Coz.  3dr.  lsc.  19gr.  to  grain-.  .  3099gr. 

4.  Redact  I  tins  to  pounds,  ouiv 

21b.  9oz.  2dr. 

5.  \l  tmce  876943  grains  to  higher  denominations. 
<".   Bednee  27.  ,  lr.  2sc  L5gr.  to  grains. 

7.    How  many  pounds,  ounce-,  etc.,  of  medicine  will  an  apoth- 
ecary use  in  preparing  974  prescriptions  of  15  grains  each? 

Ans.  21b.  6oz.  3dr.  lsc  lOgr. 

AVOTBDUPOIS  WEIGHT. 

1H>.    AyontDUPOn  WUQl  1  for  weighing  the  coarser 

articles  of  DK  h  afl  bay,  eotton,  tea,  sugar,  copper, 


iron,  etc. 

TABU. 

16  Drams  (dr.) 

make          1   Ounce, 

OZ. 

16  Oqdc 

l    Pound, 

lb. 

2.'»   ; 

l  Quarter, 

qr. 

•l  < 

1   HundredWei^ht,  cut. 

Weig 

1  Ton, 

t. 

oz. 

dr. 

lb.              1 

=                 16 

qr. 

1     =           16 

= 

iwt.                   1 

=          27,     =          400 

=         G400 

t.                1=4 

=       100     =       1C00 

=      25600 

1     =     20     =     80 

=     2000     =     32000 

=     512000 

■ 

voirdnpoU  Weight  | 

RE1'  05 

m  formerly  to  consider  281b.  a  quarter,  1 12 lb.  a 
hundred  weight  and  UU-iOlb.  a  tou  ;  but  now  the  twu<i/ practice  il  in  I 

•  different  tons  arc  i!  i  as  the  long  or  gross  ton  =  22401b. 

and  the  short  or  net  ton  =  200011). 

-till  need  in  tba  irholmVl  eot]  trade,  also  in  estimating 
goods  at  tlio  I  .  .v  .  BStOflB  !• 

und   in  Avoirdupois   "Weight   is  equal  to  7000  grains  in 
.  lit. 

Ex.  2.    In    135951b.    how 
many  tout, 

Ol'KRATIOX. 

2  5)1  35  9  51b. 

4)5  4  3qr.    -f  201b. 
2  0)  18  5cwt.  +  3gr. 
6t.      +  15cwt. 


1.    Reduce     Gt. 
ftqr.  20 lb.  to  pounds. 
opki:  mo*. 

6t.  loewt.  3qr. 
20 

15c^ 
201b. 

13  5 cut. 
4 

5  43qr. 
25 

2  73  5 
1086 

13  5  9  5  lb.,  Ans. 

Ans.  6t.  15cwt.  3qr.  201b. 


3.  Reduce  3t.  Gewt.  2qr.  51b.  6oz.  lOdr.  to  drams. 

Ans.  1703786dr. 

4.  Reduce  3G42897  drams  to  higher  denominations. 

Ans.  7t.  2cwt  lqr.  51b.  loz.  ldr. 

5.  Reduce  S7t  19cwt  oqr.  to  pounds. 

6.  Reduce  1779Glb.  to  higher  denominations. 

7.  Reduce  St  19cwt.  3qr.  241b.  15oz.  15dr.  to  drams. 

8.  K'  duoe  1 7  1268  I  drams  to  higher  denominations. 

In  lOt  lewt  2qr.  101b.,  net  weight,  how  many  gross  tons? 

10.  If  a  bone  eata  221b.  of  hay  in  one  day,  how  many  tons 
will  he  eat  in  865  da  .  -It.  Oewt  lqr.  51b. 

11.  If  a  blacksmith  uses  231b.  8oz.  of  iron  daily,  how  many 
tons  will  he  use  in  818  days? 

00.  How  many  pounds  now  make  a  ton?  How  many  formerly  ?  What  arc  the 
different  tons  called  I  for  irhftl  is  the  long  ton  now  used?  One  pound  Avoirdu- 
pois equals  how  many  grains  1 , 


Otj  REDUCTION. 

CLOTH  MEASURE. 

07.    Cloth  Measure  is  used  in  measuring  cloths,  ribbon* 
braids,  etc. 

TABLE. 

2\  Inches  (in.)  make  1  Xail,  na. 

i  Nails  ■  1  Quarter,       qr. 

4  Quarters  "  1  Yard,  yd. 

na.  in. 

qr.  1  =  2\ 

vd.  1         =         4  =  9 

1  =         4         =       1G  =  36 

xpressions  like  \,  |,  etc.,  are  called  fractions.     £  =  one  fourth 
|  *  two  thirds  ;   2$  =  two  and  one  fourth.     The  principles  of  fractions  wi] 
cussed  in  another  place. 

1.   Reduce    15yd.   3qr.         Ex.2.  In  254  nails  how  man; 
-iki.  to  nails.  yards,  quarters,  and  nails? 

OPERATION.  OPERATION. 


15  yd.  3qr. 

2na. 

4)  2  5  4na. 

4 

6  3qr. 
4 

4  )  6  3  qr.  +  2na. 
1  5  yd.  +  3qr. 

.  15yd.  3qr. 

3.  In  27yd.  2qr.  Sna.  how  Ans.  1 1:5. 

•1.   In  873  nails  how  many  yards,  etc  ?     A::-.  6  Lyd  2qr.  lna. 

5.  Hbw  many  dresses  may  be  Blade  from  107yd.  3qr.  of  silk 
if  each  dreai  requires  15yd.  lcjr.  ?  Ans.  11. 

6.  If  2yd.  3qr.  of  ribbon  are  used  in  trimming  one  bonnet 
how  many  yard*  will  be  used  in  trimming  5  bonnets? 

7.  Reduce  43yd.  2qr.  Sua.  to  nails. 

8.  It"  2yd.  iqr.  of  cloth  are  required  for  making  one  coat,  hov 
many  yards  will  be  used  in  making  8  coa: 

9.  What  cost  25yd.  3qr.  of  cloth  at  $2  per  quarter  ? 
10.  Reduce  782-4  nails  to  yards. 

97.   For  what  is  Cloth  Measure  used!    Table?    Scale?    Note! 


. 


07 


LON*.    MKASUM-:. 

9S.    Long   Mkam  u  I  In  ■MMoring  distance?,  i.  e. 

;ih  is  required  without  regard  to  breadlh  or  thickness. 

TABLE. 


rns  (p.  c.)  : 

12    \                                " 

1   Inch, 

l  Pool, 

in. 

3    F                               " 

1    Yard, 

yd. 

Yards  or  1G£  Feet     " 

l  Bod, 

rd. 

Rods                          " 

1   Furlong, 

fur. 

8    Furlongs                      " 

1  Mile, 

m. 

l.|r>      '                                            « 

1  League, 

1. 

G0^  Statute  miles,  nearly, " 

1  Degree  on  Circ.  of  the  Earth,  1° 

3 GO  Degr                        " 

1  Circumference, 

circ. 

in. 
ft.                       1     as 

b.  c. 
3 

yd. 

I    =        12    = 

36 

rd.                   1 

=        3    =z        3G    = 

108 

fur.                  1     =          t>\ 

—       16J=      198    = 

594 

m.        l    =    40    =    220 

—     6G0    =    7920    = 

237G0 

1    =z  8    =  320    =  17G0 

=±  5280    =  63360    = 

190080 

:  1.     The  earth  not  being  an  exact  sphere,  the  distance  round  it  iu 
different  directions  is  not  exactly  the  same     By  the  most  exact  measure- 
ments made,  ■  degTM  is  a  little  less  than  69£  miles. 
Note  2.     The  barleycorn  is  but  little  used. 

I  3.     The  3  before  miles  iu  the  table  is  not  a  part  of  the  scale. 

Ex.  1.  How  many  rods  in         Ex.  2.  Reduce      2710rd.     to 


8m.  3fur.  30rd.  ? 

higher  denominations. 

OILKATI 

OPERATION. 

8  m.  3fur.  30rd. 

40)27  1  Ord. 

8 

6  7  fur. 
40 

8  )  6  7  fur.  +  30rd. 
8  m.  -|-3fur. 

J710  rd.,  Ans. 

Ans.  8m.  3 fur.  30rd. 

3.  In  1yd.  feft.  8in.  how 

many  barleycorns  ?            Ans.  528. 

08.    I  Long  Measure   used?    Table?    Scale?    A  degree  upon  the 

earth,  how  long? 


68 


II  EDUCTION. 


4.  Reduce  473b.c.  to  higher  denominations. 

5.  The  distance  through  the  center  of  the  earth  is  about  7912 
mil.-  ;   how  many  rod-  is  it? 

6.  Thfl  distance  round  the  earth  is  about  8000000  rods;  how 
many  miles  is  it? 

CHAIN    MEASURE. 

99.     Chain  MbASUBI  \t  used  by  engineers  and  surveyors  ir 
measuring  roads,  canals,  boundaries  of  fields,  etc 

TABLE. 


7tVa  Inches  (in.)         make 


1   Link,  li. 


25 
4 

10 
8 

Lii. 
Rods 
Chains 
Furlongs 

1   Rod,  Perch,  or  Pole,  rd. 
1  Chain,                     ch. 

1    Furlong,                    fur. 
1  Mile,                           m. 
li.                       in. 
1     =        7flfc 
=        25     =         198 
=       100    = 
=    1000    =      : 
=    8000     =     63360 

m. 

1 

fur. 
1      = 

ch. 

1 

10 

80 

rd. 

1 

=        4 

=      40 

=    320 

Note.     To  measure  roads,  etc.,  engineers  often  use  a  chain  100  feet  long 


l \.  i.    Reduce  5m. 

8ch.  3rd.  1511  to  links. 


7fur. 


•2.  Seduce  47890  links  tc 
higher  denomination-. 


OPKRATI' 

5  m.  7fur.  8ch.  3rd. 

15  li. 

on 
2  5  )  4  7  8  9  0  li. 

8 

4  7  fur. 
10 

47  8ch. 
4 

4)  1  9  15rd.   +  1511 

10)47  8ch.    +  3rd. 

8  )  4  7  fur.  -f-  8  ch. 

.">  m.   -j-  7  fur. 

I915rd. 

Ans.  5m.  7fur.  8ch.  3rd.  15  li 

95  90 
3830 

4  7  8  9  0  li.. 

Ans. 

09.  For  abatis  Chain  Measure  used?    Table?    Scale?    Note? 


tion. 


r,o 


8.  Jn  fiftir.  2rh.  8r&  1811  bow  many  links  ?         An*.   <-293. 
•1.    Bt  dace  8879  links  to  bigbet  denominations. 

4ns.   3fur.  8ch.  3rd.  Hi. 
; M  17m.  Slur.  5dfc  2nl.  2  Hi.  t<>  link-. 
Efodace  l.">17.">  links  t<»  higher  denominations. 
Prom    Boston   to  Anduver   if   23   miles;    how  many  links 
is  it  ? 

Prom  Boston  to  Fitchburg  is  400000  links  ;    how  many 
miles  is  it  ? 

9.  The  distance  round  a  field  is  7fur.  6ch.  3rd. ;  what  will  it 
to  fenoe  the  field  at  $2  per  rod? 

10.  How  many  miles,  etc.,  in  637482  links? 

SQUARE  MEASURE. 
100.    Sqtabe  Measure  is  used  for  measuring  surf 
TABLE. 


144     Square  Inches  (sq.  in.) 

9     Square  Feet 
80]  Square  Yards  or) 

Square  Feet         j 
40     Square  Rods 

make 

u 

m 

U 

1 
1 

1 

1 

Square  Foot, 
Square  Yard, 

Square  Rod, 

Rood, 

sq.  ft. 
sq.  yd. 

sq.  rd. 

r. 

I      Roods 

m 

1 

Acre, 

a. 

f>l<>     Aeres 

u 

1 

Square  Mile, 

sq.  m. 

(a)  Also  in  Chain  Measure, 

10000  Square  Links  or) 
1  6   Square  Rods        ) 

make 

1 

Square  Chain 

,     sq.  ch. 

10  Square  Chains 

M 

1 

Acre, 

a. 

§q.  rd. 

sq.ya. 
1= 

sq.ft. 
1= 
9= 

sq.in. 

144 

1296 

1  = 

30J: 

272J= 

1=          40= 

12l0z 

10890= 

1568160 

fq.m.    1=  •    4=       100= 

10= 

43560= 

6272640 

£=$40=2560=102400=  3097600=27878400=401 44*3 

nriii^'  land,  surveyors  use  ■  4-rod  chain  composed  of  100 

.films  the  halt'-ehain  of  50  link^ 

is  mi 

loo.     I    r  what  is  Square  Mcasurt  used?     Tabl**     Scaler     Table  in  Chain 
Measure '    Note* 


70 


DEDUCTION. 


Ex.  1.  In  2sq.  m.  625a.  2r. 
25sq.  rd.  how  many  sq.  rods  ? 

OPERATION. 

2  sq.  m.  G2 5  a.  2r.  25sq.  rd. 
640 

1905a. 
4 


Ex.  B.  Reduce  804905sq.  rd, 

to  higher  denominations. 


OPERATION. 


7  ii2  2r. 
40 


40)304905  gg  rd. 

4)7  622r.  +  25sq.rd. 
6  4  0  )  1  9  0  5  a.  -f  2r. 

m.-j-('--~'.'i. 

Ans.    2>q.  m.  625a.  2r.  25sq.  rd. 
:'.  0  1  905  Sty  rd.,     An?. 

3.  In  14sq.  m.  25a.  3r.  n<Vq.  rd.  bow  many  square  rods  ? 

Ant.  l  l-;77."»0. 
1.   Redo  3sq.  rd.  to  higher  denominations. 

5.  Bought  a  field  containing  3a.  2r.  2"*sq.  rd.  at  $2  per  rod  ; 
what  did  it  cost  ? 

101.    The  manner  of  finding  the  area  of  a  surface  like  Fig. 
1,  may  be  understood  from  the  following  explanation.     Let  A  B 

represent  (on  a  reduced  scale) 


PlO.   1. 


1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

1-1 

15 

a  line  5  inches  long ;  then, 
<  vid.-mly,  if  we  pass  from  A 
to  e,  a  distance  of  1  inch,  and 
draw  the  line  e  f,  the  figure 
A  B  f  e  will  contain  5  square 
inches,  i.  e.  5  X  1  square 
inches.  So,  in  like  manner, 
D  C        A  B  h  g  will  contain  10,  or 

5X2  square  inches  ;  and 
A  B  C  D  will  contain  15,  or  5  X  3  square  inches,  i.e.  we  multi- 
ply together  the  numbers  representing  the  length  and  breadth,  and 
the  product  will  be  the  number  of  square  inches  in  the  surface. 

Note.  A  surface  like  Fig.  1  is  called  a  rectangle.  If  the  length  and 
lrcadth  are  equal,  the  rectangle  is  a  square.  The  angles  of  a  rectangle  or 
■quale  arc  all  equal  to  each  other  >  and  each  angle  is  called  a  rigid  angle. 


101.  How  ia  the  area  of  a  rectangle  or  square  ascertained?    What  is  said  of 
the  angles  of  a  rectangle  or  square?    What  is  each  angle  called? 


REDUCTION.  71 

10*2.    The  area  of  a  rectangle  divided  by  the  length  will 
tlic   breadth,  and   the  area  divided  bj  tin-  breadth  will   ghr< 

fr/iy//* ;  thus,  in  Fig.  I,  !•">  -f-  5  =  8  end  i"»  -+■  3  =  5. 

Sow  many  square  rods  in  a  field  that   ifl  7  padi  wide 
and  9  rods  long  AlH 

7.  How  many  square  rods  in  a  field  that  is  25  rode  Wide  and 
48  rods  long?     How  many  acres?  N  An-.  7a.  ft 

8.  Aboard  containing  30  square  ft-ot,  is   12  feet  long;    how 

wide  is  it  ? 

'.).  A  fiower  garden  containing  300  square  feet  is  12  feel  wide  ; 
how  long  is  it  ? 

1<>.  How  many  acres  in  afield  that  is  20  rods  wide  and  56 
>ng?  Ans.  7. 

SOLID  OR  CUBIC  MEASURE, 

103.    Solid   or   Cubic  Measure  is  used   in   measuring 
things  which  have  length,  breadth,  and  thickness. 

TABLE. 

1728  Cubic  Inches  (c.  in.)     make  1   Cubic  Foot,  cu.  ft. 

■J7   Cubic  Feet                        "  1   Cubic  Yard,  c.  yd. 

16  Cubic  Feet                        "  1  Cord  Foot,  c.  ft. 

8  Cord   Feet  or)  u  -    ^  , 

128  Cubic  Feet      /  X  ^"^  °* 

cu.  ft.  c.  in. 

p.  v.i.  1        =  1728 

1         =         27        =        46656 

Note  1.     The  scale  in  this  tabic  only  includes  1728  and  27;   the  nthpr 
numbers  arc  irregular. 

Transportation  companies  often  estimate  freight,  especially  of 

[ticks,  by  the  ipaCC  OCCOpied,  rather  than  by  the  actual  weight     In 

itimate,  from  25  or  30  to  150  or  175  cubic  feet  are  called  a  ton.      This 

i>  railed  arbitrary  weight,  and  it  varies  with  different  transportation  compa- 

:i  1  lotnewhai  according  to  the  risks  of   carriage.      The  Boston   and 

•isiders  a  thousand  of  bricks  a  ton,  whereas  tho 

BOtaal  weight  is  more  than  two  tons.  Again,  a  horse  is  estimated  at  30001b-, 

brsadtl  of  a  rectangle  found  when  the  area  and  length  are 
known'  Bow  tlic  length,  when  the  area  ami  hrendth  are  known!  103.  For 
what  is  Solid  Measure  u*e<l '    Tablet    Sea!-'    Note  2? 


72 


REDUCTION. 


though  the  average  weight  of  horses  is  not  far  from  1000  lb.  Masta,  shl| 
timber,  hard-wood  boards,  etc.,  are  estimated  at  the  rate  of  30001b.  for  41 
cubic  feet,  which  gives  2Gjj  feet  per  ton.  The  old  distinction  between  squar 
and  round  timber  is  practically  abolished.  Furniture  and  other  light  an< 
bulky  articles  are  estimated  at  150  feet  to  the  ton,  which  gives  about  8  ton 
to  a  full  freight  car-load. 

Ex.  1.    How   many  cubic         Ex.  2.    Reduce   944cu.  ft.  t< 
feet  in  34c.  yd.  26cu.  ft.?  cubic  yards  and  feet 

OPERATION.  OPERATION. 

3  4c.  yd.  2  Gcu.  ft.  2  7)  944 cu.  ft. 

->  7  ~~34c.  yd.  +  2  Gcu.  ft. 


2  04 
68 


Ans.  34c.  yd.  26cu.  ft. 
9  4  4cu.  ft.,  An<. 

3.  In  3c.  Gc.  ft.  15cu.  ft.  156c.  in.  how  many  cubic  inches  ? 

Ans.  855516. 

4.  If  40cu.  ft.  make  one  ton,  how  many  tons,  cubic  feet,  etc. 
In  889664  cubic  inches  ? 

104.    A  body  like  Fig.  2  is  called  a  prism.     Each  side,  ai 

D  FiS-  2'  C 


f 
i 
G 


E  F 

A  B  C  D  or  A  B  F  E,  is  called  a  face  of  the  prism.     If  eacl 


_    _. '_ > ' ' / 

A 

/- 

/•     ! ' 

f\    >\ 

!  ' 

A     " 

% 

v\ ) 

o 

B 

d 

b 

g 

e 

fa 

/ 

104.    What  is  a  body  like  Fig.  2  called?    What  is  one  side  of  the  prism  called 


73 

nn?le  of  the  faces  is  I  ri<jh(  9*gli  the  pri>m  is  rectangular,  and 

the  priMD  To  determine  the 

jular  prism,  first  find  the  area  of  the  upper 

A  li  C  D,  as  in  Art.  lOlj    then  going  from  A,  B,  and  C 

downward  1  inch  to  a,  1>,  and  C,  tad  pacing  a  plane  through  a,  b. 

shall  cut  off    1">   solid  inches,  i.  e.  5  X  3  X  1  solid 

inches,     So  if  a  plane  be  passed  through  d,  e,  and  f  it  will  cut 

»,  or  5  X  3  X  2  inches  etc. ;  i.  c.  the  continued  product  of 

the  numbers  expressing  the  length,  breadth,  and  depth,  will  give 

the  solid  contents  of  the  prism. 

103.  So  also,  the  solid  contents  divided  by  the  area  of  the 
top  face  will  give  the  depth;  the  contents  divided  by  the  area  of 
one  end  will  give  the  length  ;  and  the  contents  divided  by  the 
area  of  one  side  will  give  the  breadth  or  width. 

"What  are  the  solid  contents  of  Fig.  2  ? 

.  ">.  How  many  cubic  inches  in  a  rectangular  prism  or  block 
of  wood  which  is  12  inches  long,  8  inches  wide,  and  6  inches 
thick?  An,    12X8X6  =  57(3. 

G.  How  many  cubic  feet  in  a  room  which  is  18  feet  long,  15 
wide,  and  9  feet  high? 

7.  A  rectangular  block  of  marble  which  contains  96  cubic  feet, 
is  8  feet  long  and  4  feet  wide  ;  how  thick  is  it  ?     Ans.    3  feet. 

8.  A  grain-bin  which  holds  21  cubic  feet  of  grain  is  3  feet 
deep  and  2  (eel  wide;  how  long  is  it  ? 

9.  A  lady's  work-box  contains  480  cubic  inches;  it  is  12  inch- 
es long  and  5  inches  deep  |  how  wide  is  it  ? 

10.  In  a  pile  of  wood  1G  feet  long,  4  feet  wide,  and  6  feet  high, 
how  many  cord-  ?  Ans.  3. 

11.  If  a  load  of  wood  be  8  feet  long  and  4  feet  wide,  how  high 
must  it  be  to  make  a  cord? 

12.  My  bedroom    ia   1">    feet    lon«r,  12   feet  wide,  and   9   feet 
:  in  how  many  minutes  shall  I  breathe  the  room  full  of  air, 

if  1  breathe  1  cubic  foot  in  2  minutes? 

10J.     When  is  a  prism  rectangular?    When  is  it  a  cube?    How  arc  the  con- 
fa  rectangular  prism  found?     103.    How  the  depth,  length,  or  breadth,  if 
we  know  the  content*  of  tha  body  and  the  area  of  one  faco? 
7 


7  i  REDUCTION. 

LIQUID  MEASURE. 

100.  Liquid  Measure  is  used  in  measuring  all  liquids. 
The  U.  S.  Standard  Unit  of  Liquid  Measure  is  the  old  English 
wine  gallon,  which  contains  231  cubic  inch 

TABLE. 

4  Gills  (gi.)  make             1  Pint,  pt. 

2    Pints  ■                 1   Quart,  qt. 

4  Quarts  u               1  Gallon,   gal. 

pi.  gi. 

at  1        =  4 

gal.                   I',  ss  I        a  8 

1         =         4  =  8        =        32 

Note  1 .  It  has  been  customary  to  measure  milk,  and  also  beer,  ale,  and 
ntln t  malt  liquors,  by  beer  measure,  the  gallon  containing  282  cubic  inches, 
but  this  custom  is  fast  going  out  of 

Note  2.  Casks  of  various  capacities,  from  50  to  1 50  or  more  gallons, 
are  indiscriminately  called  hogsheads,  pipes,  butts,  tun-. 

Ex.  1.  In  Ggal.  3qt.  lpt.  Ex.  2.  Reduce  222  gills  t* 
2gi.  how  many  gills  ?  gallons,  quarts,  etc 

OPERATION.  OPERATION. 

ft gaL  5qt  lpt  2gL  4 )  2  2  2  gi. 

4  2)5j>pt  +2gi. 

2  7qt.  4)2  7qt.  +  1  pt. 


G  gal.  +  3  qt. 


5^Pt  Ans.   Ggal.  3qt.  lpt.  2gi. 

2  2  2gi.,  Ans, 

3.  Reduce  8gal.  2qt  lpt  3gi.  to  gilK  Ans.  279gi. 

4.  Reduce  7496  gills  to  higher  denominations. 

5.  How  many  demijohns,  each  containing  2gal.  lqt.  lpt.  3gi. 
may  be  tilled  from  a  cask  which  contains  98  gallons  and  3 
quarts? 

6.  How  many  gallons  of  molasses  in  24  jugs,  each  containing 
2gal.  3qt.  lpt? 

106.    For  what  is  Liquid  Measure  used?    Table?    Scale?    Notel?    Note  2? 


RID1  75 


DRY  MEASURE, 

107.    I>r.\  Mia -i  i;i.  i-  need  in  measuring  grain,  fruit,  pota- 

TABLE, 

I    Pints  (pt.)  in:ikc  1    Quart,  qt. 

8   Quarts  "  1    P<  pk. 

t  Peeka  "  1  Bushel,   boah. 

qt.  pt. 

pk.  1=2 

i.ush.  i       =        8       =       u; 

I        =        4        =        32        =        <;i 

.     The  bushel  ineiisnrc  is  18$  inches  In  diameter  and  8  inehc- 
nnil  contains  ft  Utile  lesfl  th:m  S150|  ■oHd  inches,  or  nearly  9J  wine  gallons. 

1.  In  Sbuah.  8pL  7qt         Ex.  2.  Reduce   2 jo  pints  to 


lpt.  how  many  pints  ? 

OPERATION. 

3bu>h.  Spk.  7qt 

lpt. 

hnnhoto  pecks,  etc. 

OPERATION. 
2)25  5})t. 

4 

15  pk. 
8 

1  2  7  qt. 
2 

8)  127qt.  +  lpt. 
4  )  1  5  pk.  +  7qt. 

3  bush.  +  8plc 
.Ajis.  3bush.  3pk.  7qt.  lpt. 

2  5  5  pt.,  Ans. 

3.  Reduce  8bush.  2pk.  3qt.  lpt.  to  pints.  Ans.  55 lpt. 

1.    Reduce  7893pt  to  higher  denominations. 

5.  Redact  4698pt  to  higher  denominations. 

8.    How  many  pints  in  15bu>h.  8pk.  6qfe  lpt.? 

7.  How  many  pints  in  2 -Ibush.  Ipk.  7qt  lpt.? 

8.  What  is  the  e  bosh.  2pk.  of  graas  seed  at  $1  n 

9.  Reduce  345G9  pints  to  higher  denominations. 
I leduce  63 bush.  2pk.  7qt.  lpt.  to  pints. 

107.     For  Wtttl  IN  Mftdf     Table?     BOfttet     What  are  the  ilinu  n. 

Of  the bushel  measure?   How  uiauy  cubic  inches  dues  it  coutaiu  '   llow  many 
*«ue  gallon*? 


76 


REDUCTION. 


TIME. 

108.  Time  is  used  in  measuring  duration.  The  natural 
divisions  of  time  are  days,  months  (moons),  seasons,  and  roars. 
The  artificial  divisions  are  seconds,  minutes,  hours,  weeks,  etc 

TABLE. 

CO  Seconds  (sec.)                         make  1  Minute,               m. 

60  Minute                                      "  1  Hour,                   h. 

24  Hours                                         «  ID                          d. 

7D                                              f  l119                    wk. 

1  W                                                   "  1  Lunar  Month,  1.  m. 

13  Months,  1  Day,  and  6  Hours   "  1  Julian  Year,  J.  yr. 

12  Calendar  Months  (=865  or)  i),  1  Civil  Year,     cyr. 

100  1                                               make  1  Century,              C. 


m. 

SCO. 

h.        1  = 

d. 

1=    60  = 

3600 

wk. 

1 

=   24=   1440  = 

l.m. 

1 

=   7 

=  168=  10080  = 

604800 

1 

=  4 

s=  28 

=  672  =  40320  = 

241. 

13Tf 

,  =  52J« 

,  =  365| 

=  8766  =  525960  = 

31557600 

J.yr. 
1  = 

Notb  1.  The  twelve  calendar  months  have  the  following  number  of 
days:  January  (Jan.)  has  31  days;  February  (Feb.),  28  (in  leap  year, 
29) ;  March  (Mar.),  31  ;  April  (Apr),  30  ;  May,  31  ;  June,  30  ;  July,  31  : 
Au_rust  (Aug.),  31  ;  September  (Sept.)i  30;  October  (Oct.),  31 ;  Xovcmbet 
(Nov.),  30  ;  December  (Dec),  31. 

Note  2.  The  number  o^  days  in  each  month  may  be  easily  remembered 
by  committing  the  following  lines  : 

"  Thirty  days  hath  September, 
April,  June,  and  November; 
All  the  rest  have  thirty-one, 
Save  the  second  month  alone, 
"Which  has  just  eight  and  a  score 
Till  leap  year  gives  it  one  more." 

Note  3.  A  solar  year,  i.  e.  a  year  by  the  sun,  is  very  nearly  365  days, 
5  hours,  48  minutes,  and  50  seconds. 

103.  For  what  is  time  used  ?  What  are  its  natural  divisions?  Artificial  divi- 
sions? Table?  Scale?  What  are  the  names  of  the  calendar  months?  How 
many  days  in  each?    Length  of  a  solar  year? 


TIOX. 


77 


Ex,  1.    Reduce    3wk. 
•m.  to  mm 

OPl  RATIO 


6d 


;!ul,  6d  23b.  59m. 
_7 

2  7d. 

_2_1 
f3  1 


J.    Reduce  40319m.   to 

orii!  crioy. 
6  0  )  4  0  3  19m. 

2  1  )  OJTl  h.  -f  59m. 
7)2  7<1.  +  23h. 

k.  +  Od. 

Ans.  3wk.  Cd.  23b.  59m. 


1  h. 
GO 

;  1  9  m.,   An>. 

:  face  lwk.  4d.  ICh.  8m.  to  minutes.         Am.  lG808m. 
1.    EKedo  seconds  to  higher  denominations. 

luce  865<L  oh.  48m.  50sec.  to  seconds. 

6.  In  •'!  \2G\)$  minutes  how  many  days,  hours,  etc.? 

7.  In  5C.  56yr.  8m.  how  many  calendar  months? 

:  'luce  37846  calendar  months  to  centuries,  years,  etc 
:  duce  2419199  seconds  to  weeks,  days,  etc. 
Reduce  34d.  20h.  40m.  50sec.  to  seconds. 


CIRCULAR  MEASURE. 

109.    Circular  Measure  is  used  in  surveying,  navigation, 
aphy,  astronomy,  etc.,  for  measuring  angles,  determining 
latitude,  longitude,  etc. 

TABLE. 


00  Seconds  (00") 
<;<)  Minutes 
80  I).  _ 

;is,  or  300° 


1     = 


i    s     II 


make         1  Minute,  1' 

1  Degree,  I9 

"             1  Sign,  s 

"  1  Circumference,  circ. 

1'    =  00" 

1°   =           GO     =  3G00 

=3      1800    =  108000 

a     21600    =  1l>9G000 


10*J.    For  wliat  is  Circular  Measure  u-  Scale? 


78 


REDUCTION. 


Note.     A  Circle  is  a  figure  bounded  by  a 
curved    line,  all  parts  of  the  curve   being 
equally  distant  from  the  center  of  the  circle, 
Th*  aaoi    is   the    curve    which 

bounds  the  circle.  An  Aft  is  any  portion 
of  the  circumference,  as  A  B  or  B  D.  An 
arc  equal  to  a  quarter  of  the  circumference, 
or  90°,  is  called  a  qmattnmU  A  luul 
line  drawn  from  the  center  to  the  circumfcr- 
asCAorCB.  A  Diameter  is  a  line 
drawn  through  the  center  and  limited  by  the  curve,  as  A  D. 

Ex.1.  How  many  second         Ex.  2.    Reduce    632931"    to 


in  5s.  20° 

;.r 

'? 

higher  denominations. 

on 

OPERATION. 

5s.  25° 

48'  54". 

60) 

6  3  2  9  3  1 

30 
175° 
60 
10548' 

60 

)10548'+54" 

|  l  7o°-f48' 
5  s.  +  2  5 

60 

Am.   5f.2504ff 

632934 

. 

;;.  Be  ;7"to  second.  Am.  1047847* 

\*  how  many  cirrumt'er  tc.  ? 

5.  Ill  o  quadrants,  10°  s'  ."/'  how  many  seconds? 

6,  Reduce  984627"  to  oaadram%  degrees,  etc. 


MISCELLANEOUS  TABLE. 


1IO.  This  table  emhracos  a  few   tnnis  in  common  use,  and 


may  be  indefinitely  extended. 


12  Single  things 

make 

1  Dozen. 

12  Doz 

.. 

1   Gross. 

. 

12  Gross 

u 

1   Great  Gross. 

• 

20  Single  things 

u 

1  Score. 

24  Sheets  of  paper 

a 

1  Quire. 

» 

20  Quires 

u 

1  Kcam. 

196  Pounds 

m 

1  Barrel  of  Flour. 

200  Pounds 

a 

1  Barrel  of  Beef  or  Pork. 

109.    What  is  ft  Circle?  Circumference?  Arc?    Quadrant?   Radius?   Diameter? 


79 

I.    How  many  dosCU    bottles  each   bottle  holding  lqt  lpt. 

will  be  sufficient  to  1m. ti!  lqt  lpt.  of  win- 

How  man.  :    papf t  in   o   nams,  18  quires,  and  23 

Ml-(    I   I.I.AM...I   -     I'.XAMll.l  S     IN     RKOICTION. 

l.  Reduce  ¥?£.  14a,  6U  8qr,  to  farthings. 

i*.  Reduce  18busa.  8pk.  T«jt .  lpt.  to  pints. 

3.   RedoOt  7t.  1  lewt.  L>qr.  121b.  8../.  (Mr.  to  drams, 

•1.   Hbw  many  tons,  etc.,  in  57  1  «'»'J2  ounces? 

5.  Reduce  1  •"> 7 7 < » IS  seconds  to  minutes,  hours,  etc. 

«',.  Reduce  84888  grain*  to  icruples,  drams  eta 

7.  Redno  bXT  to  seconds. 

8.  Reduce  3m.  5  fur.  7ch.  2rd.  20 li.  to  links. 
'.'.  Reduce  14  lb.  7oz.  15dwt  88gr.  to  grains. 

10.  Reduce  6ft  4J  .')5  IB  6gr.  to  grains. 

11.  Reduce  27)48  equate  inches  to  higher  denominations. 
IS.  Rednee  4 1 1  nails  to  quarters  and  yards. 

18.   li.  .luce  7432  farthing--  to  pence,  etc. 

14.  Reduce  18469874  drams,  Avoirdupois,  to  ounces,  etc 

15.  Reduce  54896  grains  to  pennyweights,  etc. 

16.  Rednee  ssq.  m.  25a.  3r.  84sq.  rd.  to  square  rods. 

17.  Reduce  8c  yd.  1787c  in.  to  cubic  inch 

18.  Reduce  4sq.  yds.  to  Bquare  Inches. 

19.  Reduce  Igal.  lpt.  to  gills. 

20.  Reduce  2wk.  6d.  8h.  lCsec.  to  seconds. 

21.  Reduce  4m.  7mr.  39rd.  to  rods. 
L,L,.   Reduce  3795  rods  to  furlongs,  etc. 
23.  Reduce  17yd.  2qr.  3na.  to  nails. 

Reduce  10881  links  to  miles,  furlongs,  etc 
.  Reduce  6598  pints  to  quarts,  pecks,  etc 

Reduce  4868294"  to  higher  denominations. 
87.  Reduce  4680  gills  to  higher  denominations. 
28.  Reduce  195261  cubic  inches  to  feet  and  yards. 

Reduce  310556  square  rods  to  rood-,  acres,  and  miles. 

i:.     This  subject  will  receive  further  attention  in  the  atti<  lea  on  Frao- 
tious. 


80  DEFINITIONS   AST)  LAL    PRINCIPLES. 


DEFIXITIOX8  AND  GENERAL  PRINCIPLES. 

111.  All  numbers  are  even  or  odd. 

An  Kvi:n  NuMBJtB  is  a  number  that  is  divisible  by  2  (Alt 
71);  as  2,4,8,  It 

An  Odd  Number  is  a  number  that  is  not  divisible  by  2 ;  as  1, 
3,  5,  11,  1.'. 

112.  All  number*  are  prime  or  composite. 

A  Pkimi:   Ni  m:  number  that  is  divisible  by  no  whole 

number  except  its,  r/  as  1,  2,  3,  5,  7,  11,  19. 

Nou  1  Tuo  is  the  only  even  prime  number,  for  all  even  numbers  art 
divisible  by  2. 

Xotk  t,  Two  numbers  arc  mutually  prime  (i.  e.  prime  to  each  other)  when 
no  whole  number  but  one  will  divide  each  of  them  ;  thus,  8  und  9  are  rau- 
tually  prime,  although  neither  8  nor  9  is  absolutely  prii: 

A  (  iMFOfi]  i  I  NiMiiKR  is  a  number  (Art.  Gl)  that  is  divisible 
by  other  nun.  i   0M|  thus,  G  is  oomp 

il  is  divisible  br  3  and  b/3j  1-  b  compotito»  bncigm 

is  OOBlpO  iuse  it  is 

divisible  by  .">  and  ."). 

Note  3.     A  composite  BMtbtf  that  is  composed  of  any  number  of  equal 

factors  is  called  a  power,  and  t  ;.ic  tailed  the  roots  of  the 

.\  huh   canals  3  X  3  is  the  ser  otul  power  or  square  of  3,  and 

3  is  the  second  or  square  root  of  9  ;  CI,  wh;  4X4X4,  is  the  third 

power  or  cube  of  4,  and  4  is  the  third  or  cube  root  of  64. 

;  |  4.  The  power  of  a  number  is  usually  indicated  by  a  figure,  called 
an  index  or  exponent,  placed  at  the  n\>ht  and  a  little  al>ove  tho  number;  thus, 
the  second  power  or  square  of  4  is  wriucn  4*,  which  equals  4  X  4  =  16;  tho 
tit ird  power  or  cube  of  4  is  48,  which  equals  4  X  4  X  4  =  64. 

Note  5.     A  rtxf  may  be  indicated  by  the  radical  tig*,  -J  ;  thus,  «/9  indi- 
cates the  second  or  square  root  of  9,  which  is  3.     So  8>/8  indicates  tin-  third 
or  cube  root  of  8,  which  is  2.     The  s-fare  root  of  a  number  is  one  of  its  tico 
nutors ;  the  cu!>e  root  is  one  of  die  three  equal  factors  of  the  number. 
I  6.     Every  number  is  both  thejir>  I  thejirst  root  of  V 

111.    What  is  an  Even  Number?    An  Odd  Number?     113.  A  Prime  Number? 
What  is  the  only  even  prime   number?      When  are   numbers  mutually  prime? 
:  is  a  Composite  Numbi  :         .  f  I  '    How  is  a  power  indicated? 

A  root?    A  number  is  what  power  of  itself  ?    What  root? 


PUB.  81 

roinra  N 

113.    The  I\\<  ;  number  are  those  numbers  u1 

continued  product  u  the  number;  thus,  8  and  7  are  the  factors 

Of   J  1  ;    o  and  6,  or  3,  8,  :i:id  8  ;iiv  the  factors  of  18;   etc 

:v  iiuiiiImt  is  I  t'.u  t..r  of  itself,  the  other  fm-tor  being  I. 

Th«   prim   mdon  of  a  number  are   those  prime  numbers 
continued  product  is  the  number  ;   thus,  the    prime   fecton 
Of  12  ;nv  2,  2,  end  8 ;  the  prime  f'aetors  of  3C  are  2,  2,  3,  and 
tc 

Note  2.     Since  1,  as  a  factor,  is  useless,  it  is  not  hero  enumerated. 

19  1.  To  factor  a  Dumber  is  to  resolve  or  separate  it  into  its 
meters.     Id  resolving  a  Dumber  into  its  factors, 

The  following  facts  will   be  found  convenient : 

(a)  Every  number  whose  unit  figure  is  0,  or  an  even  number, 

u,  and  .*.  divisible  by  2. 

(b)  Any  number  is  di\  isilde  by  3  when  the  smn  of  its  digits 
(Art.  7  )  is  divisible  by  3  ;  thus,  4257  is  divisible  by  3  because 
the  .Mini  of  it>  digits,  4  +  2  +  5  +  7  =  18,  is  divisible  by  3. 

(e)    Any  UQmber  is  divisible  by   1  when    1  will  divide  the  num- 
qprODBOd  DJ  me  tl/BO  righl-Itaud  figures ;  thus,  4  will  divide 
oL\  .-.  it  will  divide  lo.Vl. 

(d)  Any  number  whose  unit  figure  is  0  or  5  is  divisible  by  5 ; 
I,  1740,  85,  8  1075,  etc. 

(e)  Any  eri  n  number  which  is  divisible  by  3  is  also  divisible 
by  C;  tin:  divisible  by  3  and  .*.  by  G. 

Note  1.    For  7  no  general  rule  id  known. 

(f)  Any  number  is  divisible  by  8  when  8  will  divide  the  num- 

by  the  three  rigid-hand  fgures  ;  thus,  8  will  divide 
.  .  it  will  divid- 

113.    What  are  the  Factors  of  a  number?    Is  a  number  a  factor  of  itself* 

What  are  \)w  prime  factors  of  a  number?     114.  What  is  it  to  factor  a  num- 

H    4?  0?    61     YVhut  is  said  of  7/ 
I          .  aiubor  ia  divisible-  ' 


82 


DEFINITIONS    AND    <.iL.NJiiL.VL   PRINCIPLES. 


(g)  An)'  number  is  divisible  by  9  when  tlie  nM  of  its  digits 
is  divisible  by  9 ;  thai,  7140  is  divisible  by  B  the  sum 

digit*,  7  +  1  +  4  +  6  =  IS,  u  divisible  by  9. 

(h)  Any  Dumber  ending  with  0  is  divisible  by  LO. 

(i)  Any  number  is  divisible  by  1 1  when  the  sum  of  the  digits 
in  the  odd  places  is  equal  to  the  sum  of  the  digits  in  the  even 
place- ;  llso  when  the  difference  of  these  sinus  is  divisible  by 
11;  thus,  8129,  in  which  9  +  1  =  2  +  8,  is  divisible  by  11; 
also  5280714,  in  which  the  sum  of  the  digits  in  the  odd  places, 
4  +  7  +  8+  6,  differs  from  the  sum  of  the  digits  in  the  even 
places,  1  +  0  +  2,  by  22,  a  number  divisible  by  11. 

(j)  Any  number  divisible  by  3  and  also  by  4,  is  divisible  by 
12  ;  and,  generally,  any  number  that  is  divisible  by  each  of  several 
numbers  that  are  mutually  prime,  is  divisible  by  the  product  of 
those  numbers;   i:  divisible  by  2,  3,  and  7,  separately, 

and  .*.  84  is  divisible  by  2  X  3  X  7  -   108  is  divis- 

ible by  4  tad  9,  and  .-.  by  4  x  9  = 

Note  2.  Every  jtrime  number,  but  2  and  5,  has  1,  3,  7,  or  9  for  its  unit 
figure. 

Por  further  aid  in   determining   t!  ..;  of  numbers,  we 

the  follow 

TABLE  OF  TRIMi:   NUMBERS   FBOM  1   TO  997. 


1 

41 

101 

167 

313 

467 

j r 

911 

2 

103 

17:1. 

57J  647 

919 

3 

L07 

i;.' 

25 1 

577 

929 

5 

109 

181 

L19 

751 

937 

7 

113 

191 

421 

757 

853 

941 

11 

Gl 

127 

193 

349 

l.il 

599 

701 

857 

947 

13 

G7 

131 

197 

271 

3,33 

859 

17 

71 

137 

L>77 

359 

521 

863 

19 

73 

211 

281 

613 

691 

787 

877 

23 

149 

283 

373 

541 

617 

701 

881 

29 

83 

151 

227 

293 

457 

017 

G19 

883 

983 

31 

89 

307 

383 

4G1 

557 

C31 

719  811 

887 

991 

37 

97 

233  311 

389 

4G3 

5G3 

641 

727  821 

907 

997 

H4.    \That  number  is  divisible  by  8?     By  10?    11?    12!     General  principle? 


i  fanaarsM,  83 

llt*i.  A  m  ii  something  to  be  domj  or,  it  U  ■  • 

tion  which  requ  Sutisfiof  ■   problem  con- 

■  •?'  tin-    operations  nece^ary  for  finding   the    answer   to  the 

question.    To  anVi  ■  problen  i  the  operations  for 

findio  s*er. 

110.    PbOBLKK  1.     To  resolve  or  separate  a  number 
its  prime  factors : 

fa  the  given  number  by  ang  prime  number  greater 

titan  one,  that  will  </,'  ,'iriile  the   QUOTIENT  hy  <nt>j  prime 

number  greater  than  one  that  will  divide  it,  oati  m  wi  ftfl  //<« 
nl  m  prime.     The  several  divieart  and  /<«;  quotient  will  be 
the  prime  factors  taught, 

.  1.   What  arc  the  prime  factors  of  30?     Ans.    2,  3,  and  5, 
on 

It  is   immaterial  in  what  order  the  prime  fac- 
o  \  j  I  tors  are  taken,  though  it   will  usually  be  most 

convenient  to  take  the  smaller  factors  first 
5 

2.  What  are  the  prime  factors  of  24?    Ans.  2,  2,  2,  and  8L 

."..  Resohre  M  into  its  prime  factors.       Ans.  2,  2,  3,  and  7. 

4.  Resolve  375  into  its  prime  factors.     Ans.  3,  5, 5,  and  5. 

5.  What  an'  the  prime  factors  of  34C5  ? 

•  "..  What  an-  the  prime  factors  of  19800? 

7  \\  :     •  are  MM  prime  factors  of  1  140? 

8.  What  arc  the  prime  factors  of  3150? 

9.  What  are  the  prime  factors  of  2310? 
Id.  What  arc  the  prime  factors  of  1728? 

11.  What  are  the  prim.-  fedora  of  1800? 

12.  What  arc  the  prime  factors  of  2448? 
L&  What  are  the  prime  factors  of  4824? 
1  1.  What  are  the  prime  factors  of  3048? 
16,  What  are  the  prime  factors  of  8696? 

16.  What  are  the  prime  factors  of  72G4? 

17.  What  are  the  prime  factors  of  507.3  ? 

115.     What  is  a  Problem'     The  snlutim  of  &  problem?     What  is  it  to  $olv*  » 
problem*     11G.   Jwule  for  fiuuiiitf  th«  prim*  factors  of  a  b umber f 


84  IIMTIUNS    AND    GENERAL    I'KINCII'LES. 

117.  If  a  number  has  composite  factors,  they  may  be  found 
by  multiplying  together   two  or  more  of  it-  pf  -s;  thu<, 

the  prime  factors  of  12  are  2,  2,  and  ;j.  and  the  composite  factors 

of  12  are   8  X  2,  2  X  8,  and  2  X  2  X  ^,  i.  c.  the  composite  fac- 
tors of  12  are   1,  6,  and  12. 

CKKATEST   COMMON  DIVISOR. 

UK  A  Common  Divisor  of  two  or  more  numbers  is  any 
number  that  trill  (/icicle  each  of  them  without  remainder ;  thus 
8  is  a  common  divisor  of  12,  18,  and  30. 

119.  The  Greatest  Common  Divisor  of  two  or  more 
numbers  is  the  greatrst  number  that  will  di\  ide  each  of  them 
without  remainder;  thus  6  is  the  greatest  common  divisor  of  12, 
18,  and  30. 

Note.     A  divisor  of  a  number  is   often  called  a  measure  of  the  number, 
also  an  aliquot  part  of  the  number. 

120.  PBOBUU  2.  To  find  tb  -t  common  divi- 
sor of  two  or             Lumbers. 

Ex.  1.    What  il  ;  lIUBOa  dfrrioorof  18,  80,  and  48? 

km.  2x3  =  6. 

KW.  We  see  that  2  and  3  are 

18  =  2X3X3  fact-  to    all    the 

30  =  2x3x5  numbers,  and,  furthermore, 

48  =  2x3x2x2x2        the?  ire  die  only  common 

factors;    hen<-e    their    prod- 
2x3  =  0,  is  the  greatest  common  divisor  of  the  gives 

numb'  r  -. 

2.  What  is  the  greatest  common  divisor  of  GO,  72,  48,  and  84? 

Ana.  2X  2x8  =  12. 

orEn.\TioK.  Although    2    is    a    factor 

GO  =  2  X  2  X  3  X  5  more  t/ian  twice  rn  some  of 

72  =  2X2X2X3X3         the  given  numbers,  y 
48  =  2  X  2  X  2  X  2  X  3         it  is  a  factor  only  twice  in 
84  =  2X2X3X7  others,  we  are  not  at  liberty 

to  take  2  more  than  twice 

117.    Composite  factors,  how  formed?     118.    What  is  a  Common  Divisor? 
110.    Greatest  Common  Divisor?    Other  Banff  for  divisor/ 


DEFINITIONS   I  ffBRAL    PRDICBFUHI.  85 

in  finding  the  |  mm  divisor.     The  same  remark  np- 

j.lio  to  ofher  metoru,     Heme, 

i  1.  nVfOJM  todl  numher  into  its  prime  factors.  and  the 
continue! product  of  all  the  prime  factors  that  are  common  to  all 
the  given  numbers  will  be  the  common  divisor  sought. 

3.  What  i>  the  greatest  common  divisor  of  24,  40,  64,  80,  9G, 
Lf0,and  i  Am.  2X  2  X  2  =  8. 

1.  Find  the  greatest  common  divisor  of  15,  4."),  75,  105,  135, 
150,  and  MO.  Ana.  15. 

Find  the  greatest  common  divisor  of  25,  45,  and  70. 

Ana.  5. 
C.  Find  the  greatest  common  divisor  of  24,  3G,  and  (5 1. 

Am-.    1. 

7.  Find  the  >mmon  divisor  of  24,  48,  72,  and  88. 

8.  Find  the  .lninon  divisor  of  45,  75,  90,  lo\"),  150, 
and  180. 

'.>.  I  have  three  rooms,  the  first  lift.  3in.  wide,  the  second 
15ft.  9in.  Wide,  and  the  third  18ft.  wide;  how  wide  is  the  widot 
carpeting    which    will    exactly    fit    each    room?        II<>w    many 

idtha  will  be  required  to  cover  each  room? 

1st  An*.  27  inches. 

191.  When  the  given  numbers  are  not  readily  resolved  into 
their  prim.'  factors,  their  greatest  common  divisor  may  be  more 
ea-ilv  found  by 

Kri.r.  -.  Divide  the  greater  of  two  numhers  If)  the  less,  and, 
if  there  be  a  remainder,  divide  the  divisor  In/  the  remainder,  and 
continue  dividing  the  last  divisor  by  the  last  remainder  indil  not/t- 
ing remains ;  the  last  divisor  is  the  greatest  common  divisor  of 
the  two  numbers. 

If  more  than  two  numbers  are  given,  find  the  greatest  dfa 

.  \hen  of  this  divisor  and  a  third  uomher.  and  so  on 
until  all  the  numbers  have  been  taken  ;  the  last  divisor  will  be  the 
■r  sought. 

Kir  lindingthe  greatest  common  divi.«or  of  two  or  more  number** 
181.  Second  rul«  for  finding  gieakst  common  divisor? 

8 


86  DEFINITIONS   AND    GENERAL   PRINCIPE: 

10.  What  is  the  greatest  common  divisor  of  11  and  20? 

OPERATION. 

14)20(1 
14 

6)14(2 
12 

Ans.      2)6(3 

"o 

Before  explaining  this  operation,  four  principles  may  be 
stated,  viz.  : 

(a)  Every  number  is  a  divisor  of  itself,  the  quotient  being 
one  ;  thus,  3  is  contained  in  3  once;  7  in  7  once. 

(b)  If  one  number  divides  another,  the  1st  will  divide  any 
number  of  times  the  2d;  thus  since  3  divides  12,  it  will  divide 
5  times  12,  or  ant/  number  of  times  12. 

(c)  If  a  Dumber  divides  each  of  two  numbers,  it  will  divide 

their  sum  and   also   tl  <  ncc ;  thus,  since    G  is   contained 

Jive  times  in  80,  and  twice  in  12,  it  is  contained  5  — |—  2  =  7  times 
in  30  +  12  =  12  ;  and  .">  —  2  =  3  times  in  30  —  12  =  18. 

(<1)   Not  only  will  the  common  divisor  of  two  numbers 

divide  their  difierenee,  but  unless  one  of  the  numbers  is  a  diviSOf 
Of  th  divide  what  remains  after  one  of  the  num- 

bers has  been  taken  from  the  other  as  many  tim«  liblej 

thus,  the  grantee!  divisor  of  6  and  22  will  divide  22  —  3  x  6  =  4. 

12*}.  It  may  now  be  shown,  1st,  that  2  is  a  common  divisor 
of  1  \  and  20,  and  2d,  that    it  i>  their  (jreatest  common  divisor. 

First,  2  divides  6,  .-.  (Art.  121,  b)  2  divides  6  X  2=12,  and 
(Art.  121.  e)  2  divides  2  +  12  =  14;  again,  since  2  divides  6 
and  1-1  (Art.  121,  e)  it  divides  6  -f-  14  =  20  ;  i.  e.  2  divides  boih 
1  1  and  20. 

Second,  The  greatest  divisor  of  14  and  20  (Art.  121,  c)  must 
divide  20  —  14  =  6,  .\  it  cannot  be  greater  than  6;  again,  the 
greatest  divisor  of  6  and  14   (Art.  121,  d)    must  divide  14  — 

181.  First  principle?    Second?    Third?    Fourth?     123.  Explain  why  2  is  a 
common  divisor  of  14  and  20.    Why  it  is  their  greatest  common  divisor. 


:.ni:kal  i-kincii'LES. 


87 


G  X  2  s=  2,  .'.  :'  :nmon  divisor  of*  14  and  20  COSUUH 

exceed  2,  ami,  ai  it  ha-  been  i»i«-\  iou-1  v  shown  that  2  is  a  divi.-or 
ot'  1  J  a:i.l  20,  ft  U  (**• r  (jreatest  conn/ion  i/irisor. 

imilar  explanation  b  applicable  in  all  oi 

123.  It  will  be  seen  that,  in  finding  the  common  divisor  of 
1  20,  \\i  are  led  to  lind  the  divisor  of  G  and  14,  then  of  2 
;  i.  e.  in  any  example  we  geek   to  fiml  the  measure  of  the 

remainder  ami  divisor,  then  of  tin-  next  remainder  and  divisor, 

0  on,  until  the  greatest  measure  of  the  last  remainder  ai  '1 

the  divisor  which  gave  that  remainder  is  found,  and  this  measure 

will  he  the  greatest  common  divisor  of  the  two  given  numbers. 

Thus  the  question  becomes  more  and  more  simple  a-  each  suc- 

tep  in  the  operation  is  taken. 

11.  "What  is  the  greatest  common  divisor  of  34.32  and  4760  ? 

opehatiox.  The  plan  of  the  operation 

in  Ex.  10  requires  more 
and   more  time   than  this  in 
1 1,  though  the  principle 
and    the    reasoning   are    pre- 
cisely the  same  in  both. 

In    Kx.  1  1    we   first   divide. 

4760  by  8432,  and  obtain  1 

for  quotient  and  1328  for 
remainder;  then  divide 

by  1328,  obtaining  2  for  quo- 
tient, and  77G  for  remainder ; 
and   so   proceed,  dividing  the 

last  divisor  by  the   lasi 
mainder,  as  directed  in  Rule 

2,  until    the    remainder    : 
The    hivt    divisor,   8,   is   the 
common  divisor  of  3432   and   47G0. 

12.  What  If  the  greatest  common  divisor  of  1430  and  3549? 

Ans.  13. 

13.  What  is  the  greatest  common  divisor  of  3G40  and  5733  ? 
1  I.   What  is  the  greatest  common  divisor  of  1440  and  86 

What  is  the  greatest  common  divisor  of  2520  and  G2J7  ? 


2656 

Quotients. 

X  1  = 

=  2  X 
X  1  = 

=  1  X 
X2  = 

=  2X 
X  6  = 

=  2X 
■ 

4760 
3432 

1328 
776 

77G 

552 

448 

208 

104 

96 

8 

1G 
1G 

0 

193.    Explain  the  opuratiwu  lu  Ex.  11. 


88  DEFINITIONS   AND    GENERAL    PRINCIPLES. 

16.  What  is  the  greaftaal  common  divisor  of  1G,  21,  and  3G  ? 

mm    on  i:a  riON.  8ECOND   OPERATION. 

J  6  )2  1  (1  24)8  6(1 

l  B  24 

8)16(9  12)24(2 

]  6  2  1 


Again,    8)3G(4  .In,     12)1G(1 

32  1^ 

Ans.  4)8(2  Ans.  4  )  12(3 

&  il 

0  0 

In  wiring  Kw  I6j  We  tirM  find  the  divi-or  of  16  and  24,  viz. 

I  then   find  the  divi-ur  of  8  and  3G  ;  or  first   Bud  the  di- 

risor  of   24  and  12, and   t  1i*-ti  < >t*  12  Hid  16  j  <>r  we 

might  firrt  find  the  divisor  of  16  and  3G,  and  then  of  that  divi- 

■nd  2  i. 
17.   What  is  the  grottoat  common  divisor  of  84,  9G,  111.  and 

1  S.    What  i>  the  gWHiOBt  common  divisor  of  77,  105,  and  140? 
19.    What  is  the  greatest  common  divisor  of  9  and  16? 

Ans.  1. 
What  if  Um  groataat  common  divisor  of  9,  12,  and  20? 

LEAST  COMMON  MULTIPLE. 

1*2  1.  A  Multiple  of  a  number  is  any  number  which  is 
divttible  by  that  number;  thus,  1">  is  a  multiple  of  5  and  also 
of  3 ;  21  is  a  multiple  of  7  and  of  3. 

Note.     Every  number  is  both  a  divisor  and  a  multiple  of  itself. 

125.  A  Common  Multiple  of  two  or  more  numbers,  is 
any  number  which  is  divisible  by  each  of  the  given  numbers ; 
thus.  48  is  a  common  multiple  of  4,  G,  and  8. 

193.  How  is  Ex.  16  solved  ?  124.  What  is  a  Multiple  of  a  number? 
125.    A  Common  Multiple  of  two  or  more  numbers? 


B&AL   PRIN0IP1 

126.     The  Li.ast  Common  IfUtTtFLl  of  two  or  mmv  mun- 
is the  least   number   that    II  divisible   by  .  adi  of  the    . 
number-;   thttS,  2  1  if  the  least  common  multiple  of  -1,  C,  and  8. 

n.     There  is  no  such  thing  as  a  least  common  divisor,  or  g] 
in  multiple. 

1*J7.    Problem  3.     To  find   the  least  common  mul- 
tiple of  two  or  more  numbers. 

1.  What  is  the  least  common  multiple  of  20,  24,  and  36  ? 
An-.  2X2X2X3X3X5  =  360. 
operation.  Since  3 GO  contains   all  the 

20  =  2  X  2  X  5  factors  of  26;  84,  lad  86,  re- 

24  =  2X2X2X3         spectively,  it,  evidently,  is  di- 
36  =  2x2x3X3        visible  bv  each  of  those  num- 
bers.    It  is  also  evident  that 

no  number  less  than  360  will  contain  20,  24,  and  36,  tor  if  one 
Of  the  2's  in  the  common  multiple  were  omitted,  it  would  not 
contain  21  ;  if  one  of  the  3's,  it  would  not  contain  30;  and  if 
the  5  were  omitted,  it  would  not  contain  20. 

Similar  reasoning  applies  in  all  examples.     Hence, 

Kile  1.    Resolve  each  number  into  its  prime  factors,  and  the 
nied  product  of  all  the  different  prime  factors,  each  taken 
the  greatest  number  of  times  it  occurs  i/i  either  of  the  given  num- 
bers, will  be  the  least  common  multiple. 

2.  What  II  the  least  common  multiple  of  12,  16,  20,  and  30? 

Ans.  2X2X2X2X3X5  =  240. 

3.  What  is  the  least  common  multiple  of  22,  33,  and  55? 

4.  What  ia  the  least  common  multiple  of  16,  36,  40,  and  48? 
What  ifl  the  teas!  common  multiple  of  20,  30,  50,  and  80? 
What  is  the  least  common  multiple  of  15,  25,  45,  and 

7.  What  is  the  least  common  multiple  of  85,  50,  75,  and 

8.  What  is  the  least  common  multiple  of  21,  36,  48,  and  64  ? 

:>.   What  is  the  least  common  multiple  of  72,  80,  81,  and  96? 
10.  What  is  the  least  common  multiple  of  42,  49,  72,  and  88? 

The  Least  Common  Multiple?  May  numbers  have  a  least  common  diet' 
visor?  Greatest  common  multiple.'  1SV*  liulc  for  finding  the  least  common 
multiple?    Keasou? 

8» 


90  DEFINITIONS    AND  GENERAL   PRINCIPLES. 

1£8.    The  same  result  is  sometimes  more  easily  attained  by 

Rule  2.     Having  set  the  given  numbers  in  a  line,  divide  by 
any  runn:  uumbtr  (hat  will  divide  two  or  more  of  them,  and  set 
tin'  quotients  and  undivided  numbers  in  a  line  beneath  ;  proceed 
with  this  line  as  with  the  first,  and  so  continue  until  no  tico  of  the 
fog  cam  be  divided  by  any  number  greater  than  one ;  the  con- 
tinued product  of  the  divisors  and  numbers  in  the  last  line  will  be 
illiple  sought. 
The  second  rule  may  be  illustrated  by  the  example  already 
employed  in  explaining  the  first  rule,  viz.: 

What  i.s  the  least  common  multiple  of  20,  24,  and  36? 

Ans.  2X2X3X0X2X3  =  360. 
operation.  If  the   process   by  the    1st  rule  be 

examined  it  will  be  seen  that  the  fac- 

"'  )  fo    1~2    To*  ,"r   ~    's    *u,m,l    '    ,mi,'*i   m    ,M<-'    .-'Vt  n 

- - imm'  !    M    8    i-    taken    but    3 

o )     5,      6,      0  time-  in  finding  the  multiple,  it  i|  re- 

5?       2,      3  jeeled   1  times.      l>y  the  2d  rule,  a! 

i-  rejected  1  times,  viz.  twice  in  the 
1st  division  by  2  and  twiee  in  the  2d  di\i>ion  by  2.  The  learner 
may  think  ted  thrcf  times  in  each  of  the  two  fir-t  <li\ laOM, 

but  he  niu-t  remember  that  the  divisar,  2,  is  retained  as  a  factor 
in  the  common  multiple  in  each  instance. 

Similar  rem  applicable  to  all  reject. -d  factors  in  like 

examples,  .-.  the  two  rules  fJLYt  the  same  result. 

1 1 .  Whet  is  the  leant  common  multiple  of  5,  1 6,  24,  32,  and  48  ? 
Ans.  2X2X2X2X2X3  X  5  =  480. 

OPERATI 

By  Rule  1.  P>y  link  2. 

5  =  5*  8  )  1,  1  6,  2  4,  3  2,  4  8 

1U  =  2X2X2X2  B   8,12,16,24 

LM  =  2X2X2X3  ;  '   '   '   ' 

^2X2X2X2X2  £ >  5>  4'  b'  °»  1  l 

48  =  2X2X2X2X3  2 )  5,  2,  3,  4,  6 

3)5,  1,  3,  2,  3 

5,  1,  1,  2,  1 

»^  . — . 

138.    Second  Rule?    Explanation? 


i.UAL    l'KIX<  Il'LES.  91 

;  in.  iplo,  wliich  Ls  tho  samo  in  tho  two  rules,  is  most  rcudily 
.ved  by  the  BrM  operation. 

What  i*  the  least  common  multiple  of  30,  40,  45,  ami  75? 
18.   What  i-  the  smallest  Rffl  of  money  with  which  I  can  buy 
ach,  cow-  /h,  QT  iheep  at  $8  each,  using 

tame  sum  in  each  case?  Ans.  $G00. 

1 1.  1  have  t  wine  measures;  the  first  bolda  1  quarts,  the 

and  5  quarts,  the  third  6  quarts,  and  the  fourth  8  quarts  ;  what 

of  the  smallest  cask  that  can  be  exactly  mea-und  by 
means  of  each  of  these  measui  -.  120  quarts. 

|5.  What  is  the  least  common  multiple  of  10,  15,  45,  75,  and 

In    BoWlDg    Ex.    15,   it  is   evident   tliat    10,    15,  and    45    may 

at   on  nek   OUt,   for    each   of   these   numbers   is   a    in<  a- 

l  .-.   whatever  multiple  of  75,  and  90  is  found,  it, 

inly,  mint  be  a  multiple  of   10,  15,  and  45;    hence,  the 

question  fo  reduoed  to  this:  What  ■  the  least  common  multiple 

of  75  ami  90? 

Many  other  abbreviations  of  this  and  other  rules  m:iv  be  pffhcSXll, 
but  a  delicate  perception  of  the  relations  of  numbers,  and  a  skillful  npplica- 
tlOfl  of  principles,  will  much  mere  faeilitate  the  progress  of  the  learner  than 
uux  let  of  formul  rules. 

(a)  If  the  numbers  are  prime,  or  even  mutually  prime,  their 
product  is  their  least  common  multiple. 

10.  What  is  the  least  common  multiple  of  9  and  10? 

Ans.  9  X  10  =  90. 

17.  What  is  the  least  common  multiple  of  8,  9,  and  25  ? 

(b)  The  least  common  multiple  of  two  numbers  is  equal  to 
their  product  divided  by  their  greatest  common  divisor. 

18.  What  is  the  least  common  multiple  of  12  and  20? 
The  amnion  divisor  of  12  and  20  is  4,  ami 
The  least  common  multiple  is  12  X  20  -f-  4  =  GO,  Ans. 

19.  AVhat  b  the  least  common  multiple  of  G3  and  72? 

<  onunon  multiple  of  33  and  77  ? 

w  oolvcd?     What  of  other  abbreviation*'      locust  common 
multiple  of  mutually  prime  numbers?    Of  tico  numbcra? 


!>-  COMMON    rKACTIOMl 


COMMON    FRACTIONS. 

129.  A  Fraction  is  an  expression  representing  one  01 
more  of  the  equal  parti  of  a  unit. 

Note.     A  unit,  or  any  other  whole  number,  is  often  called  an  Integer; 
>  called  an  Integral  or  Entire  dumber. 

130.  A  Comox  or  Vuloajb  Fraction  is  expressed  by 

two  Homo  above  and  the  other  below  a  line;  thus  ^  (one 

half),  |  (two  fifths),  etc 

(a)  The  number  below  the  line  shows  into  how  many  parts 
the  unit  is  divided,  and  is  called  the  Denominator,  because  it 

ninates  or  gives  name  to  the  part>  ;  thus,  if  a  unit  is  di\ 
into  3  equal  part-,  each  part  is  one  third;  if  into  8,  each  part  is 
one  eighth  ;  etc? 

(It)  The  number  above  the  line  is  called  the  Numerator, 
rates  or  numbers  the  parts  tc 

(<•)  The  numerator  and  the  denominator  are  the  Terms  of 
the  fraction. 

131.  A  fraction  is  nothing  more  nor  less  than  uncr- 
l  division,  i.  e.  division  indicated  but  not  performed,  the 

fwmerator  being  the  di  tnd  the  tfar  the 

divisor.     Ilence, 

(a)  The  value  of  a  fraction  is  the  quotient  of  the  nu- 
merator, divided  by  the  denominator  ;  thus,  ^  =  12  -f-  4 
=  3  :  and.  .-.. 

(b)  Any  change  in  the  numerator  causes  a  like  change 
in  the  value  of  the  fraction,  and  any  change  in  the  denomi- 
nator causes  an  OPPOSITE  change  in  the  value  of  the  fraction 
(Art.  84). 

These  principles  are  developed  in  the  following  Problems. 

129.  What  is  a  Fraction?  Other  names  for  a  whole  number?  130.  A  Com- 
mon  Fraction,  how  expressed?  Number  below  the  line,  what  called?  Why? 
Number  above,  what  called?  Why?  Terms  of  a  fraction,  what?  131.  A  frac- 
tion, what  is  it?     Value  of  a  fractiou?     What  follows? 


com  93 

132.  A  Proper  Pa  \<  HOI  b  OM  "hose  numerator  is  less 
than  the  denominator;  a-.  v.  fo  fa. 

133.  An  [MPBOPBB  I  kaction  is  one  whose  numerator 
f/u, ils  or  exceeds  its  denominator  j  a-,  ;},  i.  >,,  '{'].  An  improper 
fraction    equals  or   exceeds  a   unit;    hence    its   name,  DfPSOPEB 

Hon. 
13  I.     A  SrjfPLl   I  i:  W  riOH  has  but  one  numerator  and  one 

tinator,  an  1  i<  either /jrojoer  or  improper 
13»>.     A  ( loXPOl  "N"i>    1'i:\<ik»n   is  a  fraction  of  a  fraction  ; 
*  *  of  ft. 

136.  A  Mixr.D  NuifBSB  is  a  whole  number  and  a  fraction 
united  JOg. 

137.  A  Complex  Fraction  is  one  that  has  a  fraction  or  a 

34-    $  3 
1  number  for  one  or  for  each  of  its  terms;  as,  — ,  -,  — -, 

7     6  2% 

JL  * 

138.  Tlie  Reciprocal  of  a  number  is  a  fraction  w 
numerator  i-   1,  and  whose  denominator  is  the  number  itself; 

thus,  the  reciprocals  of  4,  9,  and  $  are  £,  $,  and  -. 

7 

Problem   1. 

139.  To  reduce  a  mixed  number  to  an  improper 
fraction. 

1.  In  3L  how  many  fourths?  Ans.  -\/\ 

on  Since  4  fourths  make  a  unit,  there  will 

be  4  times  M  many  fourths  as  units,  there- 

l  fore,  in  three  units  then-  will  be   t  times 

~J2     j  3  fourths  =12  fourth*,  and  the  1  fourth  in 

the  example  added  to  the  12  fourths, 

II  rice, 

132.     A   PrOf    r  Fraction,  what?      133.     An    hapropW   Traction?      134.    A 
— fmwt    Fraction?      130.     A  .inner? 

;  lex  Fraction?    1.18.  The  Reciprocal  of  a  Number?    130.   Explain 
raUon  in  Ex.  1. 


94  COW  \CTIONS. 

Rule.  Multiply  the  whole  number  by  the  denominator  of  the 
fraction;  to  the  product  add  the  numerator,  and  under  the  sum 
write  the  denominator. 


2.  In  5}  how  many 

sevenths  ?                                Ans. 

3.  In  8}  how  many 

little?                                         Ans. 

4.  In  $7  £  how  many  fourths  of  a  dollar? 

5.  Reduce  6$  to  an 

improper  fraction.                      Ans. 

6.  Reduce  95  to  an 

improper  faction.                    Ans. 

V- 

7.  Reduce  5g. 

17.  Bedaoe  19^. 

&  Reduce  9|. 

18.  Reduce  16|. 

B  done  Uf 

1.'.   K.  .luce  20?. 

10.  Reduce  13|. 

90,  Bedi 

11.  Redact  U 

H,  Reduce  37§. 

Li,  Reduce  15|. 

22.  Reduce  46  ,V 

13.  Bedaoe  17|. 

23.  Reduce  54&. 

14.  Reduce  11 

24.  Reduce  84^. 

Reduce  168V 

25.  Reduce  99 

1C.  Reduce  18g. 

20.  Reduce  99jj|. 

(a)  To  reduce  an  integer  to  a  fraction  having  any 
i  denominator: 

Multiply  the  integer  by  the  proposed  denominator,  and  under 
the  product  write  tfie  denominator  (Art.  84,  c). 
87,  Reduce  19  to  a  fraction  wl.  tnmatnr  is  7. 

28.  Reduce  9  to  a  fraction  whose  denominator  is  8. 
9  to  a  fraction  \vh  ninator  is  •"». 

30.  Bedttoe  7  to  a  faction  whose  denominator  is  1. 

Ans.    }. 

31.  Reduce  87  to  a  fraction  whose  denominator  is  87. 
.   Reduce  10  to  a  fraction  whose  denominator  is  1. 

luce  16  to  a  fraction  whose  denominator  is  4. 

34.  Reduce  20  to  a  fraction  whose  denominator  is  4. 

35.  Reduce  14  to  five  different  fractional  forms. 

139.  Rule  for  reducing  a  mixed  number  to  an  improper  fraction?    ReasonT 
a«  integer,  how  reduced  to  a  fractional  form? 


COMM<»\    IftAOZBOJ 

Problem  2. 

1 10.  To  reduce  an  Improper  fraction  to  a  whole  or 
mixed  Dumber. 

1.   How  many  units  in  Y  ?  Ana.    8|« 

Since  the  mmieratoT  is  i 

^=r  13  H- 4  =  3|,     A  dividend  and    tin-  d<ni>mina- 

lof  ■  drriior  (Art.  l.si ).  the 
reduced  to  an  equivalent  whole  or  mixed   number  by 
the  following 

Kite.  Divide  the  numerator  by  the  denominator  ;  if  there  is 
any  remainder,  place  it  over  the  divisor,  and  annex  the  fraction 
so  formed  to  the  quot 

%   Etednoe  (|  tO  I  wholo  or  mixed  number.  Ans.    3,1,. 

Bedooe  H  t<>  ■  whole  or  mixed  number.  Ans.  3. 

4.  Reduce  JJ^  to  a  whole  or  mixed  number.  Ans.    2}f. 

5.  I               _.•;*•  to  a  whole  or  mixed  number.  Ans.    26g*ff. 

9.  Reduce  *£. 

7.  Reduce  f\  10.  Reduce  §|. 

8.  1  11.  Reduce  i£f&. 

Problem  "-. 
141.    To  reduce  a  fraction  to  its  lowest  terms. 

!.   Reduce  j|  to  its  lowest  terms.  Ans.    J. 

Dividing  both  terms  of  a  frao 

run  omunov.  tionby  any  number  does  not  alt  ei 

3|  =  }}  =  3,    Ans.  the  value  of  the  fraction  (Art.  84, 

b,   and    131) ;    .'.   dividing  ead| 

term  of  ||  by  8  Lrivrs  the  equal  fraction  M ;  then  dividing  eacjb 

term  of  this  remit  by  1  gaVee  ,\  and  m  3  and  4  are  mutually 

prune  (Art  1  12),  |§,  in  it-  bweet  terms,  equals  f. 

In  this  operation  both  terms  of 
bkcov  :iok.  the  traction  ||  are  divided  by  their 

12)5§  =  2>      An<.  itest    common    divisor,  19  (Art. 

119),  and  thus  the   fraction  is  re- 
duced :tt  oner  tO  LtS  lowest  tflfflDS*      11cm 

i  to.    Rule  for  reducing  an  improper  fraction  to  a  whole  or  mixed  number? 


96  COMMON   FRACTIONS. 

Rule  1.  Divide  each  term  by  any  factor  common  to  themj 
then  divide  these  quotients  by  any  factor  common  to  them,  and  so 
proceed  till  the  quotients  are  mutually  prime.     Or, 

Rule  2.     Divide  each  term  by  their  greatest  common  divisor. 

2.  Reduce  §£  to  its  lowest  terms.  Ans.    §. 

3.  Reduce  $£  to  its  lowest  terms.  Ans.    |. 

4.  Reduce  fa  to  its  lowest  terms.  Ans.    \. 

5.  Reduce  tW&  to  li3  lowest  terms.  Ans.    ft. 
G.  Reduce^.                           11.  Reduce  i£ft. 

7.  Reduce  $?.  12.  Reduce  ffo. 

8.  Reduce  |  13.  Reduce  §§*. 

9.  Reduce  £f £.  1  L.  Induce  ^%. 
10.  Reduce  gj}$.  15.  Reduce  |||. 

Problem    1. 
142.    To  multiply  a  fraction  by  a  whole  number. 
Ex.  1.  Multiply  ft  by  3.  Ans.  ft  or  £. 

It  is  just  as  evident  that   3 
first  operation,  times  ft  are  ft  as  that  3  times  2 

ft  X  3  =  ft,    Ans.  cents  are  6  cents,  or  that  3  timet  S 

are  6  ;  i.  e.  when  the  numerator 
is  multiplied  by  3  the  fraction  represents  3  times  as  many  parts 
as  before,  and  each  part  continues  of  the  same  size ;  .*.  the  frac- 
tion is  multiplied  by  3. 

If  the  denominator  is  divided 

secon-d  operation.  by  3,  the  fraction  represents  just 

ft  X  3  =  $,    Ans.  as  many  parts  as  before,  but  each 

part  is  three  times  as  great,  and 

.-.  the  whole  fraction  is  three  times  as  great.     Hence, 

Rule  1.     Jfultiply  the  numerator  by  the  whole  number.     Or, 
Rule  2.     Divide  the  denominator  by  the  whole  number. 

Note  1.  The  correctness  of  Rule  1  is  also  evident  from  Art.  83  (a),  and 
Art.  131.     Rule  2  also  depends  on  Art.  83  (d).  ■ 

141.  First  rule  for  reducing  a  fraction  to  its  lowest  terms?  Second  rule? 
RatMNfcl  142.  First  rule  for  multiplying  a  fraction  by  a  whole  number?  Why? 
Second  rule?    Why?    Another  reason ? 


COMMON    FRA<  CIONS.  97 

2.  Multiply  T\  bj  3.  Ans.  fa  or  $. 

ferable  in  this  and  all  similar  examples, 
because  it  ghrti  tho  faction  In  smaller  terms. 

3.  Multiply  475  by  5.  Ans.  $. 

4.  Multiply  i\  by  11.  Ans.  f  or  1?. 

5.  Multiply  yV  by  4. 

13tX4  — ^,  by  Rulel;  or, 

VVX4  =  |-,byRule2. 

Note  3.    The  first  rule  is  preferable  for  this  and  all  similar  examples,  ba- 
cause  the  second  gives  a  complex  fraction. 

6.  Multiply  ft  by  4.  Ans.  |f  or  — . 

7.  Multiply  rf>  by  6.  Ans.  $?. 

8.  Multiply  ^T  by  4.  Ans.  §$. 

9.  Multiply  fc$  by  3.  Ans.  §f. 

10.  Multiply  1$  by  5.  Ans.  f|. 

11.  Multiply  ft  by  4. 

12.  Multiply  fs  by  5. 

.   Multiply  rttrby  15.  Ans.  ^V 

14  Multiply  ^  by  15.  15  =  5  X  3. 

A  X  5  =  f  ;  and  f  X  3  =  V,  Ans. 

i  4.  We  may  here,  ns  in  whole  numbers  (Art.  61 ),  use  the  factors  of 
the  multiplier,  and  in  using  these  factors  we  may  apply  the  1st  or  the  2d 
rule,  or  both. 

15.  Multiply  |i  by  G6.  66  =  6  X  H- 

«  X  6  =  t| ;  and  «  X  11  =  W>  Ans. 
1G.  Multiply  ||  by  42.  Ans.  i|i. 

17.  Multiply  v6\  by  84. 
Multiply  iff  by  44. 

(a)  If  we  multiply  a  fraction  by  its  denominator,  the 
product  will  be  the  numerator. 

19.  Multiply  |  by  8.  Ans.  J  X  8  =  {  =  7,  by  Rule  2. 

20.  Multiply  ||  l.y  44 

14».  May  the  factors  of  the  multiplier  be  used?    What  is  the  product  if  a  frac- 
tion is  multiplied  by  its  denominator? 

9 


08  COMMON    FRACTIONS. 

(b)  To  multiply  a  mixed  number  by  an  integer : 

Multiply  the  fractional  part  and  the  entire  part  separately,  and 
add  the  products  together ;  or,  reduce  the  mixed  number  to  an 
improper  fraction  (Art.  139),  and  tJien  nmlti]>ly. 

21.  Multiply  3$  by  5-  Am.  19. 
First  multiply  $  by  5  and  the  product  is  4 ;  then  multiply  3 

by  5  and  the  product  is  15.  These  partial  products  added  give 
15_|_4— 19  for  the  true  product.  Or,  first  reduce  3|  to  *£ 
and  then  multiply  by  5  and  the  product  is  19,  as  before. 

22.  Multiply  8?  by  9. 

$X9  =  3£;8X9  =  72;  tind  72  +  8f  =  75$.,  Ant 

23.  Multiply  9W  by  12.  Am.  113t\. 

24.  Multiply  18|  by  20. 

25.  Multiply  23 £  by  7. 

Problem  5. 

143.    To  divide  a  fraction  by  a  whole  number. 
Ex.  1.  Divide  §  by  4.  Am.  jj  or  ^. 

It  is  just  as  evident  that  one  fourth 
first  operation         of  |  is  §  as  that  one  fourth  of  8  cents 
|  -i-  4  =  § ,  Ans.         is  2  cents,  or  that  one  fourth  of  8  is  2  ; 
i.  e.  when  the  numerator  is  divided  by 
4  the  fraction  represents  only  one  fourth  M  many  porta  as  be- 
fore, and  each  part  continues  of  the  same  size;  .-.  the  fraction 
is  divided  by  4. 

If  the  denominator  is  multipled 

second  operation.  by  4,  the  fraction  represents  just  as 

f  -T-  4  =  ^r,  Ans.  many  parts  as  before,  but  each  part 

is  only  one  fourth  as  greaty  and  .*, 

the  whole  fraction  is  only  one  fourth  as  great.     Hence, 

Rule  1.     Divide  the  numerator  by  the  whole  number.     Or, 

Rule  2.     Multiply  the  denominator  by  the  whole  number. 

Note  1.     These  rules  may  also  be  explained  by  Art.  83  (b)  and  (c). 

143.  How  is  a  mixed  number  multiplied  by  an  integer?  Another  way? 
143.  First  rule  for  dividing  a  fraction  by  a  whole  number?  "Why?  Second 
rule?    Why?    Another  explanation? 


COMM'»\  99 

2.  Divide  \}  by  2.        Ans.  W  hy  Rule  1  I  hi  by  ™*  2. 

initio.     Why? 

i,   Divide  1 1  by  6.  Ans.  2V 

1.  Divide  H  hY  n- 
Divide  H  1}y  25- 

j  by  12. 
7.  Divide  tf  by  4. 

H-M  =  ^|,byRulel;  or, 

Note  3.     Tlic  2d  rule  is  preferable  in  this  example.     Why  T 
B,    Divide  \l  by  5.  An?.  ^. 

9.  Divide  fig  by  11.  Ans.  fft- 

10.  Divide  ]','  by  6. 

11.  Divide  3$  by  4. 

12.  Divide  &  by  20.  20  =  4  X  5. 

&-*-*  =  A,  and  ft  -f-  5  =  T*y,  Ans. 
4.     See  Art  142,  Note  4. 

13.  Divide  \§  by  85.  35  =  5  X  7. 

M* *=*V  ■■*  A-*-7=  y?x»  Amu 
1  1.  Divide  $J  by  18.  Ans.  W 

15.  Divide  ^  by  14.  Ans.  fo. 

Divide  a?±  by  44. 

i  To  divide  a  mixed  number  by  a  whole  number. 

17.   Divide  23$  by  1.  Ans.  5f. 

Fir-t  divide  as  in  Art.  71, 
4  )  8  Ex.  35,  and  obtain  the  quo- 

Qoo,T7. .  3*  Rem.  10i,'nf'  'I;  a»«l  ,'1"  remainder, 

31.     Then  reduce  3£  to  the 

3 J  =  V»  a11*1  V  +  4  =  f,        improper  fraction,  V-,  divide 

,•.  5-f-f  =  -r>.',.  An*  it  by  4,  and  add  or  annex  tlie 

raanltj  |,  to  the  partial  quo- 

5,  and  we  have  .">$  for  the  true  quotient, 

143.    May  tin-  factor-  <>f  Hm  dlllWt  be  used  separately?     A  mixed  number, 
now  divided  by  an  integer? 


100  COMMON  FRACTIONS. 

18.  Divide  27$  by  6.  Ans.  4f. 

19.  Divide  17g  by  9.  Ans.  1|J. 

20.  Divide  65^  by  8. 

21.  Divide  5|  by  7. 

5i  =  4£;  ^--7  =  fi,Ans. 

Note  5.    In  Ex.  21,  the  dividend  is  less  than  the  divisor;  hence  the  quo- 
tient is  a  proper  fraction. 

22.  Divide  7^  by  9.  Ans.  f  $. 

23.  Divide  5f  by  11. 

24.  Divide  $6|  equally  between  9  boys. 

Problem  6. 

144.    To  multiply  a  fraction  by  a  fraction. 

Ex.  1.  Multiply  $  by  $.  Ans.  &. 

To  multiply  $  by  £,  1st,  *  X  3  =  f  (Art  142,  Rule  1)  ;  but 
the  multiplier,  3,  is  5  times  $,  .*.  the  product,  $,  is  5  times  the 
product  sought;  hence,  2d,  f  -?-  5  =  &  (Art.  143,  Rule  2)  is 
the  product  sought ;  i.  e. 

?X&  =  &.     Hence, 

Rule.  Multiply  the  numerators  together  for  a  new  numerator, 
and  the  denominators  for  a  new  denominator. 

2.  Multiply  y\  by  $.  An?.  ^. 

3.  Multiply  &  by  J.  Ans. 

4.  Multiply  f'by  Jz.  Ans.  AV 

5.  Multiply  \%  by  -ft. 

6.  Multiply  tf  by  £$. 

(a)  To  multiply  by  a  fraction  is  only  to  multiply  by 
the  numerator,  and  then  divide  the  product  by  the  de- 
nominator. 

In  Ex.  7  we  multiply  £$  by  5,  and  obtain  *f-  (Art.  142,  Rule 
2),  and  then  ^  divided  by  6  gives  ?  (Art.  143,  Rule  1),  the 
result  sought. 

144.  Rule  for  multiplying  one  fraction  by  another?  Reason?  To  multiply 
by  a  fraction,  what  is  it?    What  principles  in  the  operation  in  Ex.  7? 


COMMON   FRACTIOH1.  101 

Multiply  H  by  f. 
2 

7 
In  t/tis  rimpU  operation  it  rm  wiiole  principle 

LINO.     To  i-'inrrl  (i.  <•.  ftriJb  owf,  or  reject)  any  fac- 
tor of  a  Dumber,  is  to  divide  the  number  by  the  rejected  i'aetor; 
thus  85  is  the  saint*  as  5  X  7,  and  if  the  5  is  canceled,  there  will 
remain  <>nly  7,  which  is  the  quotient  of  3o  divided  by  J. 
8.  Multiply  if  by  ft. 

4  2 

**v**_8       A 

5  9 

The  8th  example  is  solved  on  the  same  principle  as  the  7th. 

12  X  14       ,.  ,     . 

It   may   be    written    thus,   — — -,   which    is    the    same    as 

oo  X  *' 

4  y  3  y  2  y  7 

- - -,  and  then  canceling  3  and  7,  i.  e.  dividing  both 

5X7X9X3 

numerator  and  denominator  by  3  and  7  (Art.  84,  b,  and  131)  we 
_         4X2        8 
5X9       45 
Note.     There  can  be  no  difficulty  in  canceling  so  long  as  we  remember 
the  simple  principle,  that  it  rests  upon  rejecting  equal  factors  from  dividend 
and  divisor  (Art.  84,  b).     The  process  is  only  to  strike  out  or  cancel  the  same 
factors  from  numerator  and  denominator,  and  it  often  saves  much  labor.    It 
can  be  profitably  applied  whenever  the  product  of  two  or  more  numbers  ia 
to  constitute  a  dividend,  and  the  product  of  other  numbers  is  to  constitute  a 
-,  provided  that  there  are  equal  factors  in  the  dividend  and  divisor. 

Multiply  ||  by  f  $. 

2  5  In  this  example,  cancel  23 

4&       &$ 10  with  46,  giving  2  in  the  nume- 

*  <20  =  1~7'         S"  rator;  and  th.-n  cam-el  5  in  L\> 

and  85,  giving  5  in  the  numera- 
tor and  17  in  the  denominator. 

144.  Explain  Ex.  7.     On  wh»t  principle*  does  canceling  rest?    When  should 
It  U  applied? 

9* 


102  COMMON   FRACTIONS. 

10.  Multiply  f|  by  ft.  Ans.  ft. 

11.  Multiply  §|  by  §}. 

12.  Multiply  $$  by  J|. 

13.  Multiply  tfft  by  «. 

(b)  In  canceling  3  and  5  in  Example  14,  we  obtain  the 
quotients  1  and  1  in  the  numerators,  and  whenever  an  entire 
term  cancels  we  obtain  1  to  place  instead  of  the  teen  ctnodod; 
but  since  1,  as  a  multiplier  or  divisor,  is  valueless,  there  is  no 
need  of  retaining  it  under  any  circumstances,  except  where  all 
the  numerators  are  canceled  ;  in  such  a  case,  1  t*  the  true  MMMT- 
ator,  and  must  be  retained. 

14.  Multiply  ft  by  ft. 

1  1 

$         t>         1      K 

5         4 

15.  Multiply  }*t  by  ftV 

1 

m  v  n     i 


5  | 

16.  Multiply  V  by  V- 

5         4 

17.  Multiply  f  f  by  *fo. 

18.  Multiply  ||  by  ft^ 

19.  Multiply  | J  by  Jf.  Ans.  0. 

20.  Multiply  ff  by  JJ.  Ans.  f> 

21.  Multiply  ft^  by  Jf. 

(c)  To  reduce  a  compound  fraction  to  a  simple  one. 

22.  What  part  of  an  apple  is  $  of  }  of  it?  Ans.  Jj. 
If  £  of  an  apple  be  divided  into  7  equal  parts,  one  of  those 

parts  will  be  ft  of  the  whole  apple;  and  if  }  of  £  is  ft,  then  | 

144.    la  canceling  when  should  the  quotient  1  be  retained? 


COM  lOo 

of  $  will  be  ft,  and  9  of  ;»l  will  be  Hi  i.  e.  «  ampmmdfi* 

>/e  one  by  the  rult-  f>r  multijilyint/  a  frac- 

Multiply  |  by  V).  i.  a.  reduce  |  of  £  to  a  staple  frac- 
tion. 

2  I.  Redace  9  of  §  of  \$  to  a  simpl.-  fraction.       Ans.  }$g. 
Reduce  3  of  ^  of  \\  to  ■  simple  fraction. 

16,  What  is  I  of  g  of  ^  of  f  of  $  of  f  of  I  of  f  ? 

£x|xfx<x£xfx£>44*« 

27.  1  v<  (luce  §  of  f  of  -ft  of  I  to  a  simple  fraction. 

28.  Reduce  $  of  }?  of  tf  of  ft  to  a  simple  fraction. 

29.  What  co.^t  J  of  B  yard  of  cloth  at  *  of  ■  dollar  per  yard? 

An-.   I  of  a  dollar. 

30.  If  a  man   builds   $  of  a  rod  of  wall  in  a  day,  how  much 
will  he  build  in  £  of  a  day? 

31.  A  man  owning  g  of  a  farm  sold  g-  of  his  share ;  what  part 
of  the  farm  did  he  sell? 

(d)  To  multiply  ■  whole  Dumber  by  a  fraction. 

At  $8  a  barrel  what  will  ^  of  a  barrel  of  flour  < 

Aih. 
nasi  oMnunosr.  If  a  barrel  costs  $8,  then  1 

4)  $  8,  Price  of  1  bid.  fourth  of  a  barrel   will  cost  \  of 

ZTZ   „    t    c  ,  1 1  1  $8,  viz.  $2,  and  3   fourths  will 

Costof|bbl.  |2=$6>Ana. 

$  6,  Cost  of  %  bbl. 

SECOND    OI  M.KATION.  ■.'.*«.  A«  ,  r»,,» 

^  Q  ^  .        e  ..  ,  .  If  lbbl.  costs  $8,  then  3  bbl. 

$  8,  Price  of  lbbl.  ^  ^  3  ^  ^  =  §2  u  ;mil 

sincv  I  of  :;  i.i)i.  u  the  same 

4  )  $2  4,  Cost  of  3 bbl.  of  lbbl.  we   divide  the  cost  of 

~$~c,  Cost  of  \ bbi.        ■*•*  *7* :m;1  ■• ,iml  th;'.  "* 

of  ^  of  a  barred,  viz.  $6,  which  is 
imc  result  as  by  the  fir-t  operation. 

144.     How  in  a  compound   fraction  reduced  to  a  pimple  one?    How  many 
wayt  to  multiply  an  integer  by  a  fractiou  ?    First  method  f    Second  • 


104  COMMON   FRACTIONS. 

33.  Multiply  24  by  |;  i.  e.  find  £  of  24.  Ans.  15. 

34.  If  an  acre  of  land  costs  $45,  what  will  $  of  an  acre  cost? 

35.  What  is  the  value  of  £  of  a  buahfll  of  clover  seed,  hi  ^7 
per  bushel?  Ans.  $5J, 

(e)  To  multiply  a  mixed  number  by  a  fraction  or. 
mixed  number: 

Reduce  each  factor  to  the  form  of  a  fraction  and  then  multiply 
the  fractions  togeth  • 

36.  Multiply  22  by  1$. 

2|Xlt  =  V'X*  =  H  =  4iJ>  Ans. 

37.  What  cost  2|  yards  of  cloth,  at  %\\  per  yard  ? 

Ans.  $3f. 

38.  What  cost  \\  cords  of  wood,  at  $6£  per  cord  ? 

39.  How  many  square  rods  of  land  in  a  garden  that  is  G§  rods 
long  and  5 J  rods  wide? 

Problem  7. 

145.    To  divide  a  fraction  by  a  fraction. 

Ex.  1.  Divide  \  by  f.  Ans.  \%. 

To  divide  \  by  $,  1st,  §  -r-  5  =  ft  (Art.  143,  Rule  2)  ;  but 
the  divisor,  5,  is  7  times  $,  .*.  (Art.  83,  f )  the  quotient  ft  is  only 
}  of  the  quotient  sought ;  hence,  2d,  ft  X  7  =  Jf  (Art.  1 42,  Rule 
1)  is  the  quotient  sought ;  i.  e. 

1-^  =  |  Xi  =  H.     Hence, 

Rule.  Invert  the  divisor,  and  then  proceed  as  in  multiplica- 
tion (Art.  144). 

The  rule  may  be  otherwise  explained  as  follow.-  \ 
First,  To  divide  by  any  number  is  the  same  as  to  multiply  by 
its  reciprocal  (Art.  138). 

Thus,  12  -j-  4  =  3,  and  also  12  X  J  =  3. 

Again,  $  —  4  =  ^,  and  also  ^  X  i  =  ^J  i  e.  dividing  by  4 

144.   Rule  for  multiplying  a  mixed  number  by  a  mixed  number?    145.   Rula 
for  dividing  a  fraction  by  a  fraction?    Reason?    Second  explanation? 


COMMON   FRACT1<»  105 

an<l  multiplying  by  the  reciprocal  of  4,  viz.  |,  we  have  the  quo- 
jiml  to  the  product, 

'ocal  of  a  fraction  is  the  fraction  mvertod; 

thus  the  reciprocal  of  f  is  -  (Art.  138),  and,  multiplying  both 

*  1 

numerator  and  denominator  of  this  complex  fraction.  .,  by  7.  ire 

obtain  g  ;  but  multiplying  both  terms  of  a  fraction  by  the  same 

number  does  not   change  it-  value  (Art.  84,  a),  ,\  -  =  \ ;  i.e. 

the  reciprocal  of  |  ii  \  \  and,  generaUv^  the  reciprocal  of  any 
fraction  is  t/tat  fraction  inverted.  Hence,  to  divide  by  a  frac- 
tion, invert  ike  divisor  and  multiply. 

:.  Divide  |  by  ft.  Ans.  Jf  =  2^ 

;.    Divide  i  by  §. 
•J.  Divide  ft  by 

Divide  ,:,  bj  Ans.  j$$. 

G.  Divide  ft  by  fj. 

7.  Divide  9  of  J  byVof  £. 

?X^-X---X^X-^X--^   An. 
7X4'    5  X9  — *X  4X11X£  — 44'  A 

8.  Divide  g  of  £  of  £  by  §  of  f  Ans.  |f  ==  Iff 

9.  Divide  §  of  J  by  \  of  §  of  £. 
tO.   Divide  I  of  I  of  I  by  }  of  3. 

(a)  If  the  denominator  of  the  divisor  is  like  that  of  the  divi- 
dend. a<  in  Ex.  11,  they  may  both  be  disregarded  ;  for,  evidently, 
ft  are  contained  in  jji  ju-t  ai  many  times  tt  6  apples  are  con- 
tained in  21  applet,  of  6  in  24  ;  i.  e.  §f  -;-  ft  =  24  -j-  6  = 
numerator  of  dividend  -|-  numerator  of  divisor:  and  this  is 
equally  true  when  the  numerator  of  the  dividend  is  not  a 
multiple  of  the   numerator  of  the  divi>or:    thus,  £  -f-  $  =  o  -+- 

11.  Divi  ft.  Ans.    I. 

12,  Divide  ?$by  ft.  Ana,  12. 

145.    How  is  the  dhrMra  nominator*  are  alike? 


106  COMMON   FRACTIONS. 

13.  Divide  j\  by  yV  Ans.  }. 

14.  Divide  )}  by  ft. 

15.  Divide  |f  by  #f 

16.  Divide  §£§  by  f]f 

17.  Divide  m  b7  Mf  •  Ans-  HI  =  I- 

18.  Divide  |}}  by  */&. 

(b)  When  the  numerator  and  denominator  of  the  divisor  are 
respectively  factors  of  the  corresponding  terms  of  the  dividend, 
as  in  Ex.  19,  it  is  best  to  divide  numerator  by  numerator,  and 
denominator  by  denominator.  This  mode  is  true  in  all  examples 
but  not  always  convenient.     Why  true  ?     Why  not  convenient? 

19.  Divide  W  by  }  Ans.  ft. 

20.  Divide  Tf  j,  by  AV 

21.  If  |  of  a  yard  of  cloth  cost  ^  of  a  dollar,  what  costs  1 
yard  ? 

22.  If  I  earn  -ft  of  a  dollar  in  g  *f  a  day,  what  shall  I  earn 
in  ]  day  ? 

23.  If  I  pay  §  of  a  dollar  for  J  of  a  bushel  of  corn,  what 
shall  I  pay  for  1  bushel  ?  Ans.  $1£. 

(c)  To  divide  a  whole  or  mixed  number  by  a  fraction 
or  mixed  number: 

Reduce  divisor  and  dividend  each  to  the  form  of  a  simple  frac- 
tion, and  then  divide  by  the  rule  already  gic>  n. 
21.  Divide  82  by  3^. 

8}-*-3}  =  V-*-  j  =  j  =  2J,  Au<. 
25.  Divide  8  by  3  j. 

8-s-8i  =  f-5-¥  =  fX  A  =  «  =  2A»Ans. 
'26.  When  3^ lb.  of  beef  cost  43|  cents,  what  is  the  price  per 
pound?  Ans.  12£  cents. 

27.  B  traveled  19}£  miles  in  5}  hours ;  how  far  did  he  travel 
per  hour  ? 

28.  B  traveled  19} \  miles,  going  at  the  rate  of  3§  miles  per 
hour ;  how  many  hours  did  he  travel  ? 

145.    Mode  of  dividing  when  the  terms  of  the  divisor  are  factors  of  the  term* 
*>f  the  dividend?    To  divide  a  mixed  number  by  a  mixed  number? 


srs.  Iu7 

Pkoblem  8. 

110.    To  reduce  a  complex  fraction  to  a  simple  one. 

a 
1.  The  complex  fraction  |  equals  what  simple  fraction? 

operation  required  is  only  to  divide  a  fraction  by  a  frac- 
I 
tion  ;  thus,  !  =  £-j-f  =  $Xf  =  iM.     Hence, 

IJn.i:.      First,  if  necessary,  reduce  the  numerator  and  denomu 

of  the  complex  fraction   each  to  a  simple  fraction  ;  then 

the  fractional  numerator  by  the  fractional  denominator 

(Art.  11.".)." 

Note.  A  complex  fraction  may  also  be  made  simple  by  multiplying 
each  term  of  the  complex  fraction  by  the  least  common  multiple  of  their 
denominators;  thus,  in  Ex.  I,  the  least  common  multiple  of  the  two  de- 
nominators, 4  and  7,  is  88,  whose  factors  are  4  and  7.  Multiplying  the 
numerator,  J,  by  4,  gives  3  (Art.  142,  a),  and  multiplying  3  by  7,  the  other 
factor  of  the  multiple,  gives  21  for  the  numerator  of  the  reduced  fractioa 
In  like  manner,  multiplying  the  denominator,  £,  by  7,  and  that  product  by 
4,  gives  20  for  the  denominator  of  the  reduced  fraction. 

Ex.  2.  Reduce  ——  to  a  simple  fraction. 
•ft 

3.  Reduce  j—  to  a  simple  fraction.  Ans.  \. 

4.  Reduce  ^-.  8.  Reduce  -A 

5.  Reduce  *&  9.  Reduce  **♦£!*. 
G.  Reduce  •£                             10.  Reduce 


6J  4| 

$  H.  Reduce^. 


!»'''•     l  Dm  reducing  a  complex  fraction  to  a  simple  one?    Keason?    An* 

otber  mode! 


\  08  COMMON    FBAOnOHfc 

12.  Reduce  -  to  a  simple  fraction. 

7  =  $  -f-  G  =  ^,  Ans.,  by  Art.  143,  Rule  2  ;  or, 
G 

|  =  |  X  \  =  ^,  Ans.,  by  Art.  84  (a)  and  Art.  142  (a). 

13.  Reduce  ^  to  a  simple  fraction. 

g 

14.  Reduce  -  to  a  simple  fraction. 

j=l  x  §= P  Ans->  by Art- 115  (c>- 

15.  Reduce   *°f  *°f  *0f*   to  its  simplest  form.    Ans.  1. 

V  ot  t\r  ot  4  ot  I 

Problem  9. 

1  17.  To  reduce  fractions  that  have  not  a  common 
denominator  to  equivalent  fractions  that  have  a  common 
denominator. 

Kx.  1.  Reduce  %  and  (j  to  equivalent  fractions  having  a  com- 
mon denominator.  Ana,    1}  and  \'{. 

operation.  Multiplying  both  terms  of  each  fraction 

2      7       14        by  the  denominator  of  the  other  fraction 

o  X  =-=  rr  will   not    alter   the  value  of  either  fraction 

f Alt.  84,  a),  hut    it   will   necessarily  make 
r        o        i-  the  denominators   alike,  for  each   new   de- 

_  x  -  =.  —         nominator  is  the  product  of  the  two  given 
'       3       «  denominators. 

Similar  reasoning  applies,  however  many  fractions  are  to  be 
reduced.     Hence, 

Rule  1.  MultijAtj  all  the  denominators  together  for  a  common 
denominator,  and  multiply  each  numerator  into  the  continued 
product  of  all  the  denominators,  except  its  own,  for  new  numer- 
ators. 

147.    Common  denominator,  how  found  by  Rule  1?   How  the  numerators? 
Explanation? 


COMM"X    1KACTI0NS. 

2.  Reduce  J,  J,  and  £  to  equivalent  fractions  having  i  com- 
mon denominator! 

OPKllATIOX. 

4x7x9=-"-  -"*-'  common  denominator, 

8  X  7  X  0=  189,  14  numerator, 

f>X4xO=  ls,,<  M  Domerator, 

1x4x7=   28,  8d  Domerator; 
,  h  Md  |  =  }||,  igS,  and  #„  Ana, 
a  Reduce  3,  3,  and  *}.  Ans.  £fo  133,  and  ^. 

i.  Redoee  {,  3,  and  ?.  An 

5.  Reduce  §,  |,  *,  and  J.       Ans.  ^| J,  338,  kit  ■«« 
G.  Badnc  tod  ».       11.  Reduce  .*.,  |,  <.  and  1  \. 

7.  Redoee  /•,.  12.  Redoee  },  ;■.  i.  and  T7T. 

8.  Reduce  fo  t\,  and  T4j.    13.  Reduce  |,  ;,.  »,  and 

9.  Reduce  §,  $,  and  ,',.         1  L  Redoee  $\,  t>4.,  ;1',,  and 
10.  Redoee  (  l  5.  Reduce  ^,  ^,  225,  and 

(a)  The  foregoing  rule  w  ill  always  give  a  common  denomina- 
tor, but  not  always  the  least  integral  common  denominator;  this, 
ever,  may  always  be  effected  by 
Kile  2.     Reduce  each  fraction,  if  ?iecessan/,  to   itt  lowest 
(Art.  ill).    Find  ike  Uaet  common  multiple  of  the  de- 
nominatort  (An.  127)  for  a  common  denominator*    Divide  tin's 
multiple,  by  each  given  denominator*  and  multiply  tike  a 

rt  for  )/<  ir  timni  rators. 
Notk  1.     Kadi  of  tli  bonded  OB  the  principle  that  multiplying 

l>oth  terms  of  ■  fraction  by  tlie  same  numher  does  not  alter  its  value. 

l &   Redoee  |,  |,  and  17,. 

operation   i:v   Tin:   BBCOVP   1:11.1:. 

2X2X3X2=  24,  least  e.»m- 
mon  multiple  of  denominator-, 

^X3=    9,  1st  numerator, 
^X5  =  20,  2<1  Domerator, 

;    X  7  =  14,  3d  numerator; 

.-.  h|9 and  &=:&,&, end  \\,  a 

i  » 7     KuU«  for  finding  the  least  common  denominator?    Kale  for  finding  tu« 
numerators*    Principle' 

10 


a 

2)8' 

0 

6' 

V 
12 

2)4, 

3, 

6 

8)2, 

3, 

3 

110  COMMON   FRACTIONS. 

17.  Reduce  A,  f,  f,  and  |.  Ans.  |g,  f  g,  ft  Jg. 

18.  Reduce  &,  A>  Ai  and  fo- 
ld. Reduce  At  A.M.  "ndl  J- 

Kote  2.  The  first  clause  of  Rule  2  is  omitted  by  many  authors,  but  ita 
necessity  is  apparent  from  the  following  example : 

20.  Reduce  f ,  A>  and  A  to  equivalent  fractions  having  the 
least  common  denominator. 

Disregarding  the  first  clause  of  the  rule,  we  find  72  to  be  the 
least  common  multiple  of  the  denominators,  and  the  fractions  §, 
A>  and  A»  reduce  to  fa,  fa,  and  fa;  but,  regarding  the  lint 
clause,  we  have  |,  A>  and  A  =  !>  4>  and  £  =  A*  i4s>  an<1 
which  have  a  common  denominator  less  than  7 $. 

21.  Reduce  A>  |,  A.  and  $•  Ans.  $$,  $&,  J  J,  and  J  J. 

22.  Reduce  f ,  fa,  A>  ^d  A- 

23.  Reduce  A.  H»  A.  and  fa. 

24.  Reduce  iJ,  JJ,  A.  ">d  «. 

Note  3.  In  this  and  the  following  problems,  each  fraction  should  be 
in  its  simplest  form  before  applying  the  rule. 

25.  Reduce  §  of  J  and  -i 

*  5  s 

§of  |  =  |;  i^  =  ¥-f.«  =  S;  but 

§  and  ^  =  A  and  ?£,  Ans. 

26.  Reduce  J  of  f,  2J,  ^-,  and  A- 

27.  Reduce  |,  J  of  g ,  and  A* 

Remark.  The  numerators,  as  well  as  the  denominators,  of 
fractions,  may  be  made  alike  by  reduction ;  thus,  §  and  £  are 
equal  in  value  to  fa  and  \$  ;  also  4  and  A  =  H  and  H  »  a^s0 
4,  Ai  and  $■  =  |J,  31,  and  f  | ;  etc  The  process  is  simple,  but 
of  little  practical  importance,  and  therefore  seldom  presented  in 
Arithmetic. 

147.    May  the  numerators  of  fractions  be  made  alike.*    How 7 


\CTIONS.  Ill 

KI.I.U     10. 

118.  To  reduce  a  fraction  of  a  higher  denomination 
to  a  fraction  of  a  lower  denomination. 

1 .  Reduce  J  of  a  penny,  to  the  fraction  of  a  farthing. 
1  pennj  i<  equal  to  I  fiurthingt,  so  any  fraction  of  a  penny 
will  be   1  lime-  as  great  a  fraction  of  a  farthing;  .*.  £d.=  4  times 

2.  Reduce  ^  of  a  shilling  to  the  fraction  of  a  farthing. 

La  is  equal  to  12d.,  so  ^s.  =  12  times  ^d.  =  }d.,  and 
id.  =  4  times  jqr.  =  f  qr.,  Ans.     Hence, 

BULB.  Multiply  the  fraction  by  such  numbers  as  are  neces- 
sary to  reduce  the  given  to  the  required  denomination. 

3.  Reduce  Jss.  to  the  fraction  of  a  farthing. 

^s.  (=  ^d.  X  12)  =  Jd.  (  =  $qr.  X  4)  =  W-,  Ans. ;  or, 
7X12X4_7X^X4_28 
36 W~3 3  qr-'  AnS-'  **  bet0re' 

Note  1.  The  sijrn  of  multiplication,  in  these  examples,  is  written  only 
between  the  numbers  which  are  given  before  the  canceling  is  begun;  thus, 
in  Ex.  8,  M  -i-n  is  written  between  36  and  3,  for  they  are  not  to  be  multiplied 
together,  but  the  3  is  obtained  by  canceling  12  in  36.  So  in  Ex.  4,  the  12 
comes  from  fWtHffg  20  in  240,  and  the  3  from  canceling  4  in  12. 

4.  Reduce  5J5  of  a  ton  to  the  fraction  of  a  dram. 

7X20X*X25X16X16       44800 

m~  ~w~i =  "T-dr" Ans- 

5.  Reduce  \\  of  a  rod  to  the  fraction  of  a  barleycorn. 

11  6 

10  X  16*  X  12  X  3  _  10  X  33  X  tt  X  3  __  1980,         . 

-$ix  2  ~  7  b-c->An8- 

7 
n  2.     In  the  first  statement  of  Ex.  5,  the  16 J,  in  the  numerator,  is 
equal  to  ^,  and,  in  the  second  statement,  the  33  is  retained  in  the  nuuiera- 
IOC  in  the  dividend,  and  the  2  is  put  in  the  denominator  as  a  fao. 

tor  in  the  <li | 

i  »s.  Rule  for  reducing  a  fraction  iVom  a  higher  to  a  lower  denomination f 
Explanation?    How  is  Ex.  5  solved? 


112  COMMON    FRACTIONS. 

C.  Reduce  -gfo  of  a  pound,  Troy  Weight,  to  the  fraction  of  a 
grain.  Ans.  J-£&. 

7.  Reduce  ^^  of  a  pound,  Apothecaries'  Weight,  to  the  frac- 
tion of  a  grain.  Ans.  J-gS-. 

8.  Reduce  T5\j7  of  a  day  to  the  fraction  of  a  second. 

Ans.  if*. 

9.  Reduce  ^  of  a  bushel  to  the  fraction  of  a  pint. 

Ans.  iff. 

10.  Reduce  ^  of  a  gallon  to  the  fraction  of  a  gill. 

11.  Reduce  sb^c.  yd.  to  the  fraction  of  a  cubic  inch. 

12.  Reduce  J4  of  a  sign  to  the  fraction  of  a  second. 

13.  Reduce  isis^sq.  in.  to  the  fraction  of  a  rod. 

Ans.  3^*. 

1  L  Reduce  ^jfffur.  to  the  traction  of  a  link.  .  ||. 

].'>.  Reduce    .,t3  of  an  acre  to  the  fraction  of  a  square  yard, 

16.  Reduce  ^yd.  of  cloth  to  the  traction  of  as  inch. 

17.  Reduce  ^circ.  to  the  fraction  of  a  second. 

18.  Reduce  fy  of  a  ton  t<>  the  traction  of  an  ounce. 

19.  Reduce  3</W  °f  a  day  to  tne  faction  of  a  second. 

20.  Reduce  gH0«£  to  tne  fraction  of  a  farthing. 

•J  1 .   Reduce  ft  of  a  bushel  to  the  fraction  of  a  pint. 

Problem  11. 

140,    To  reduce  a  fraction  of  a  lower  denomination 
to  a  fraction  of  a  higher  denomination. 

Ex.  1.   Reduce  %  of  a  barleycorn  to  the  fraction  of  an  inch. 
In  1 "»  barleycorns  there  is  only  £  of  15  inches,  so  in  £  of  a 
barleycorn  there  is  only  J  of  §  of  an  ineh^^  of  an  inch,  Ans. 

2.  Reduce  ^  of  a  gill  to  the  fraction  of  a  quart. 

As  1  gill  is  \  of  a  pint,  so  |^gi.  is  |  of  ffpt.  =  /^pt.  and,  for 
A  like  reason,  *ypt.  is  \  of  ^\qt.  =  ^qt.,  Ans.     Hence, 

Rile.     Divide  the  given  fraction  by  such  numbers   as  are 
required  to  reduce  the  given  to  the  required  denomination. 


149.    Rule  for  reducing  a  fraction  from  a  lower  to  a  higher  denomination! 
Explanation ! 


113 

8.   Reduce  ^jftqr.  to  the  traction  <>f  a  shilling. 
Jtfqr.  (=  *£&.  +  4)  =  {d  (=  («.  ~  1 1)  —  B^8.,  Ans. ;  or, 
<gg     7  7 

18       ;;,;A„s.,  as  before. 

4.  Reduce  ±±§£&dr.  to  the  fraction  of  a  ton. 

44*00    2S00    in    7         _  7 

3  X  *0  X  ti  X  ^  X  4  X  20~240  tons'  An3* 

5.  Reduce  J-^-^b.c.  to  the  fraction  of  a  rod. 

10 

i98o _xw0    xm  m    g_io 

7X3X12X3X  5^—7  X  3  X  tft  X  3  X  XX~2ir±An5' 

f>.  Reduce  igfigr.  to  the  fraction  of  a  pound,  Apothecaries! 
ht.  Ans.  Tfar 

7.  Reduce  1§agr.  to  the  fraction  of  a  pound,  Troy  Weight. 

8.  Redu  to  the  fraction  of  a  day. 

9.  Reduce  ^in.  to  the  fraction  of  a  yard,  Cloth  Measure. 

10.  Reduce  l§&see.  to  the  fraction  of  a  week. 

11.  Reduce  ±92sfl  "i.  t0  tne  fraction  of  a  yard. 

12.  Reduce  V^  links  to  the  fraction  of  a  furlong. 

13.  Reduce  'Y^yd.  to  the  fraction  of  an  acre.        Ans.  %^v. 

14.  Reduce  Jffi  seconds  to  the  fraction  of  a  sign. 
1"».  Reduce  }  Jj  gills  to  the  fraction  of  a  gallon. 

Problem  12. 
150.   To  reduce  a  fraction  of  a  higher  denomination 
to  whole  numbers  of  lower  denominations. 

1.     Reduce  J£  to  shillings  and  pence.         Ans.  3s.  4d. 
(=  Js.  X  20)  =  i^s.  =  3 Js. ;  again  $s.  (==  $d.  X  12)  = 
4d. ;  .-.  J£r=r3s.  4d.,  Ans.     Hence, 

Krir.      Reduce  the  gi  ion  to  a  fraction  of  the  next 

lower  denomination  (Art  1  18) ;  tkem\if the  Jraetiem  is  improper, 
reduce  it  to  a  whole  or  mixed  number  (Art.  14<>).     If  the  result  is 

150.    Rule  for  reducing  a  fraction  of  a  higher  denomination   to   Integer!  o/ 
lower  denominations?    Explain.' 

10» 


114  COMMON   FRACTIONS. 

a  mixed  number,  reduce  the  fractional  part  of  it  to  the  next  lower 
denomination,  as  before,  and  so  proceed  as  far  as  desirable. 

Note.  If,  at  any  time,  the  reduced  fraction  is  proper,  there  will  be  no 
whole  number  of  that  denomination. 

2.  Reduce  %\£  to  whole  numbers  of  lower  denominations. 

J|£  (=  if  s.  X  20)  =  f  f  s.  =  4^8. ;  fas.  (=  ^d.  X  12)  = 
$d.,  a  proper  fraction ;  £  d.  (=  f  qr.  X  4)  =  3qr. ;  ,\  J  J  £  = 
4s.  Od.  3qr.,  Ans. 

3.  Reduce  fa  of  an  acre  to  lower  denominations. 

.  lr.  17rd.  18yd.  1ft.  50$in. 

4.  Reduce  fa  of  a  furlong  to  rod-,  yard. 

Ans.  J8rd.  3yd.  2ft. 

5.  Reduce  $  of  a  week  to  days. 

C.  Reduce  §|$  of  a  rod,  L  ure,  to  yards,  etc 

7.  Bedace  HI8J  °^a  cucumference  t<>  signs,  etc 

8.  Reduce  fa  of  a  ton  to  hundred  • 

9.  Reduce  £||tt>  to  oun<  <  ~.  drams,  Bcrn| 

10.  Reduce  ffflfacxrc.  to  ■ 

11.  Reduce  }}  of  a  civil  year  (865  days)  to  days,  etc 

12.  What  is  the  value  of  fa«4%  of  a  pound  Troy  ? 

13.  What  is  the  value  of  tf  of  a  but 

14.  What  is  the  value  of  £$  of  a  gallon  ? 

15.  What  is  the  value  of  fa  of  a  pound.  Apothecaries' 
Weight? 

16.  Reduce  fa  of  a  mile  to  furlongs,  chains,  etc. 

17.  Reduce  ^  of  a  cord  to  cord  feet,  cubic  feet,  etc 

18.  Reduce  fa  of  a  yard  to  quarters,  nail-,  etc. 

Problem  13. 

151.  To  reduce  whole  numbers  of  lower  denomina- 
tions to  the  fraction  of  a  higher  denomination. 

Ex.  1.  One  farthing  is  what  part  of  a  penny?  Ans.  \. 

Since  4  farthings  make  a  penny,  1  farthing  is  \  of  a  penny. 

2.  Six  pence  and  1  farthing  are  what  part  of  a  shilling  ? 
6d.  -L-  lqr.  =  25qr;  and  Is.  =  48qr. ;  .*.  6d.  and  lqr.  z=  ||s.,  Ans. 


co.v  11' 

To  determine  what  part  one  thing  is  of  another,  considered  as 

.>  or  whole  t!,in>j.  the  part  is  always  made  the  numerator  of  a 

{•on,  and  the  unit  or  whole  thi/t;/  is  j>ut  for  the  denominator  ; 

tfcos,  the  traction  -it  expresses  the   part   that  o  miles  is  of  o  miles. 

comparison  can  be  made,  the  part  and  the  whole  most 

be  of  the  same   kind  or  denomination  ;  thus,  :5  pecks  is  not  g  of 
5  bushels,  hut,  reducing  the  5  bushels  to  20  pecks,  we  hav.    :; 
-  equal  to  J'0  of  20  peeks,  i.  e.  *?$  of  5  bushels.     Benee, 

Kri.i:  1.      Unhirr  the  given  quantity  to  the  lowest  denomina- 
'  /•  a  mssurofcr ;  and  reduce  a   unit   of  the 

higher  denomination    to  the  same  denomination   as  the  numerator, 

3.  Reduce  Grd.  5ft.  9in.  to  the  fraction  of  a  furlong. 

Grd.  5ft.  9in.  =  1257in.  and  lfur.  =  7920in. 

.-.  Grd.  5ft.  9in.  =  }$S&fur-  =  sV&far.,  Ans. 

4.  Reduce  7oz.  4d\vt.  to  the  fraction  of  a  pound.        Ans.  £. 

5.  Reduce  9  rods,  1  foot,  and  6  inches  to  the  fraction  of  a 
furlong. 

9rd.  1ft.  Gin.  =  1800in.  and  lfur.  =  7920in. ; 

.-.  9rd.  1ft.  Gin.  =  ^§^fur.  =  ^ur.,  Ans. 

(a)  In  Ex.  5,  Cin.  =  ift. ;  lift.  =  JyA  =  ^rd.  and  9-rVrd. 
=  tj'Y'rd.  =  .Afur.,  Ans.,  as  by  Rule  1.     Hence, 

Ivii.i:  2.     ]>;>■■'<!,'  the  number  of  the  lowest  denomination  given 
by  the  number  required  to  reduce  it  to  the  next  higher  de?wmina~ 
vmd  annex  the  fractional  quotient  so  obtained  to  the  given 
numbtf  of  that  higher  denomination  ;   divide  the  mixed  number 
so  forme,/  Ay  the  number  required  to  reduce  it  to  the  next  higher 
>>n,  annex  the  to  the  given  number  of  that 

denomination,  and  so  proceed  as  far  as  necessary. 

1.     This  rate  is  frequently  preferable  to  the  1st,  because  it  enables 
■f,  t<>  use  smaller  numbers  and  giTSf  the  result  in  lower  terms. 

l">l.  !:  ■  lor  MdadBf  the  lower  denominations  of  a  compound  number  to 
a  fraction  of  a  higher  .1.  nomination  ?  Explanation?  Principle?  Second  rule  for 
reducing  integers  of  lower  denominations  to  the  fraction  of  a  higher  denomi- 
nation?   Explanation .'    Why  preferable  to  Rule  1? 


116  COMMON    F HAITI > 

6.  Reduce  lr.  2sq.  rd.  20sq.  yd.  laq.  it.  72aq«  in.  to  the  fraction 
of  an  acre.  Ans.  x*5. 

7.  Reduce  4oz.  Cdwt.  9ggr.  to  the  fraction  of  a  pound. 

Ana, 

Note  2.  In  Example  7,  by  Rule  1,  reduce  4oz.  6dwt.  9$gr.  to  Jifths  of 
a  grain  for  a  numerator,  and  lib.  to  fifths  of  a  grain  for  a  denominator. 
How  shall  it  be  done  by  Rule  2  ?     Which  mode  is  preferable  I     Wliv  I 

8.  Reduce  lpk.  3qt.  lpt  to  the  fraction  of  a  bushel. 

9.  Reduce  (>s.  20°  20'  30"  to  the  fraction  of  a  circumference. 
10.  Reduce  lm.  2fur.  llrd.  2yd.  lit  2}b.  c.  to  the  fraction  of 

a  league. 

J 1.  Reduce  Iqr.  2na.  ^in.  to  the  fraction  of  a  yard. 

12.  Reduce  3wk.  6d.  Uh.  27m.  to  the  fraction  of  a  Julian  year. 

13.  Reduce  lqt  lpt  lfgi  to  the  fraction  of  a  gallon. 

14.  Reduce  4  cord  feet,  12  cubic  feet,  and  1982]  cubic  inches 
to  the  fraction  of  a  cord.  Ans.  g. 

15    Reduce  3oz.  4dr.  Itc  lOgr,  to  the  fraction  of  a  pound. 
1G.  Reduce  4  fur.  6dL  Sid  20li.  to  the.  fraction  of  a  mile. 

17.  Reduce  tlcwt  1 1  lb.  loz.  12fdr.  to  the  fraction  of  a  ton. 

18.  Reduce  3  bushels,  1  peck,  4  quarts,  and  1  pint  to  the 
fraction  of  a  bushel.  Ans.  *j / . 

Note  3.  Sometimes,  as  in  Ex.  18,  the  number  called  the  part  is  greater 
than  the  unit  with  which  it  is  compare  iul  to  the  unit. 

Problem  1 1. 

lotj.  If  numbers  of  the  same  kind  are  added  together,  their 
sum  will  be  of  the  same  kind  as  the  numbers  added;  thus,  3 
books  -f-  4  books  =  7  books  ;  3  hats  -\-  4  hats  =  7  hats  ;  and 
for  a  like  reason,  £-[-$  =  £;  ^  _L-  ^  =  ^  etc.,  etc. 

(a)  Numbers  of  different  kinds  cannot  be  united  by  addition  ; 
thus,  3  hats  -f-  4  books  are  neither  7  hats  nor  7  books ;  so  £  -f~ 
%  are  neither  $  nor  $  ;  but  numbers  that  are  unlike  may  some- 
times be  made  alike  by  reduction,  and  then  added  ;  thus, 

l+t  =  tt  +  H  (Art.  147)  =  «. 

(b)  Again,  2bush.  -j- 3pk.  are  neither  5bu»h.  nor  5pkl ;  but 
2bush.  ==  Spk.,  and  then  8pk.  -f-  3pk.  =  llpk. ;    so  Shush,  -j- 


co.v  117 

i\>k.  nt  neither  }bush.  nor  5 |»k.  ;  lmt  Shush.  =  8nk.  (Art.  148), 

and  then  ;pk.  -f-  ^pk»=  Vpk-     H" 
To  vdd  fraction 

Ettn  'nee  the  fractions,  if  necessary,  fret  to  the  same 

,'natio/i,  then  to  a  common  d> .  rhich  write 

m  0/  the  new  numerators  over  the  common  denominator. 

1.  Add  (■-,  end  ,\  together,  Ans. 

kher.  Ans.  [g. 

3.  Add  Jv,  Jv,  ,'V,  and  }?  together.  Ans.  ?f  =  2ft. 

1.   Add  J    and  J  together.  Ans.  V  =  *t  =  U- 

5.  Add  ^,  ^5,  -ft,  and  ^  together.  Ans.  1$. 

C.  Add  ,.  fa  and  /:,  together. 

7.  Add  §3,  &,  ^  Hi  Mrf  A  together. 

8.  Add  together  ft,  t7*,  if,  it,  and  ft.  Ans.  2£. 
Add  together  3*ff,  Jg,  Jf,  £g,  and  ft. 

10.  Add  together  gg,  gg,  gg,  and  |g. 

1 1.  Add  together  T|F,  ft^,  ^ft,  T*w  »"<*  iW- 

1  _'.  Add  together  J,  g,  f ,  and  £.  Ana.  4$. 

13.  Add  together  gg,  *g,  *g,  and  gg. 

14.  Add  together  g,  ft,  and  ft.       -* 

§  +  ft  +  ft  =  it  +  fg  +  ig(Art.  147,Rule2)  = 
=  1*1  Ans. 
1"».   Add  together  g  and  g. 
3  +  I  =  H  +  M  (Art.  147,  Rule  1)  =  §g  =  ljg,  Ans. 

16.  Add  together  ft,  |i  tnd  $. 

17.  Add  together  ft,  ft,  and  §. 

+  A  +  i  =  i  +  i  +  3=*  =  li,Ans. 

18.  Add  .  and  gg. 

19.  Add  g  of  g  to  g  of  *g.  }  +  *  =  J,  Ans. 

20.  Add  g  of  ?g  to  g  of  jjg. 

21.  Add   'S'h  to  J  of  |.  Ans.   ft. 

to  8  x  A- 

15*.    Rule  for  adding  fraction*?    Can  nnlike  number§  be  added?    Of  what 

kind  i»  the  sum  of  two  or  more  numbers? 


118  COMMON   FRACTK' 

23.  Add  }§.  to  $d. 

|8.  +  §d.  =  Vd.  +  H=Ud.  +  i8d.  =  «d.  =  5A^Ans., 
or,  f  s.  +  §d.  =  f  s.  -f-  As.  =  fgs-  +  ^s.  =  |^s.,  2d  Ans. 
=  1st  Ans. 

24.  Add  fgal.  to  £qt.  Ans.  f  $qt.  or  fjgal. 

25.  Add  together  ^bush.  $pk.  and  £qt. 

26.  Add  together  $ton  fcwt.  and  $qr. 

(c)  To  add  two  fractions  that  have  a  common  numer- 
ator : 

Multiply  the  sum  of  the  denominators  by  either  numerator,  and 
place  the  product  over  the  product  of  the  denominators. 

27.  What  is  the  sum  of  ?  and  |  ? 

1-1-1  —  ^+1  —  15        3       3_3xl5_45 
7~i~8~~7X8~56;  •',7i"8~"      56      ~56'  AnS* 

28.  What  is  the  sum  of  £  and  £  ?  £f  =  l/n  A*19- 

29.  What  is  the  sum  of  f  and  ft  ? 

(d)  To  add  mixed  numbers : 

Add  the  sum  of  the  fractions  to  the  sum  of  the  integers. 

30.  What  is  the  sum  of  3f  and  4}  ? 

|+|=»»  +  H==tt  =  lA5  3  +  4  =  7; 

.-.  3*  +  4|  =  7  +  1 A  =  8ft,  Ans. 

31.  What  is  the  sum  of  5|,  3$,  and  12$£___JLns.  21  }f- 

32.  What  is  the  sum  of  18ft,  5|,  and  24|? 

33.  What  is  the  sum  of  15$,  24,  7*,  and  ^  ?     _ 

34.  What  is  the  sum  of  3^,  6&,  4^,  and  8 

35.  What  is  the  sum  of  £  of  $  of  6},  ^,  and  41  ? 

36.  What  is  the  sum  of  |t,  3f ,  6|,  and  J  of  J  ? 

37.  What  is  the  sum  of  3 -ft,  4&,  8^,  and  25  ? 

38.  How  many  are  8$  -f  3£  -f-  8£  +  14  ? 

153.    Mode  of  adding  two  fractions  that  have  like  numerators?     Mode  of 
adding  mixed  numbers? 


COMMON  >NS.  119 

Problem  16, 

Itl3.   To  subttf  fraction  from  a  greater  : 

r.     Prepare  the  fractions  as  in  addition,  and  then 
the  difference  of  the  numerators  over  the  common  denominator. 

1.  From  A  take  tV  A  — A  =  A=i»  An*- 

2.  From  tf  take  J7.  Ans.  J7. 

3.  From  J§  take  fc 

4.  From  £?  take  J$. 

5.  From  ||  take  J|. 
From  ^  take 

7.  Take  ft  from  £g.  An?.  ft. 

8.  Take  §$  from  |f. 
Take  }|  from  §1. 

10.  Take  fft  from  /, 

11.  From  $  take  §. 

(a)    t  —  # =H~ «  =  A.  Ans-     (See  Art-  152>  »)• 

12.  From  |  tai.  Ans.  £$. 

13.  From  ^  take  ft. 

14.  From  J  |  take  |.  « — §  =  §f  —  A  =  *  An- 

15.  From  ft  take  f.  Ans.  ^. 

16.  From  \l  take  ^. 

17.  From  ij  take  | 
LR   From  ||  take^. 

fi  —  A  =  A  —  A  =  AV  —  A1*  =  A4*  =  A.  Ans. 

19.  From  3737ff  take  ^fo.  A..<.  |J. 

20.  From  fft  take  ^. 

31.  From  J  of  |  take  £  of  *. 

|X*—*X*  =  f—f =»—»—«,  Am. 

.   From  J  of  #  take  $  of  -ft-  Ans.  ||. 

From  ?of  \$  take  j|  of  [%. 
•ji.  From  |  of  >.■;  take  ft  of 
From  A  of  &  take  &  of 

Of  |  take  $  of  vVof  §. 

I  for  subtracting  one  fraction  from  another?    Uow  are  toe  fractions 
prepared  in  addition  ' 


120  COMMON   FRACTIONS. 

27.  From  ?|  take  If.  Ans.  U. 

^  =  ^  =  a^-j-J^.  =  ^x13j  =  5;  Complex  fractions 

7  a  \     reduced   to  simple 

g|  =  *  =  V  +  ¥  =  i(ArLH5,b);j      ones. 

f  — *  =  «  —  «  =  «.  Ans. 

28.  From  ^  take  ^|.  Ans.  H  =  1 H- 

29.  From  -f  take  |. 

30.  From  }s.  take  £d.     (See  Art.  152,  b). 

(b)  |3.  — Jd.  =  Vd— Jd.  =  JJd.  — Ad.=  jfd.  =  llJd.,AM. 
or,  fs.  —  £d.  =  }s.  —  ^s.  =  ;£frs.  —  ,^8.  =  **&s.,  2d  Ans. 

31.  From  §qt.  take  £pt  Ans.  ^.(jt.  or  l£pt 

32.  From  §  ton  take  $cwt. 

33.  From  $  acre  take  f  rod.  Ans.  i|f$a.  or  67$§rd. 

Note.     The  answer  to  these  examples  may  be  in  any  denomination  of 
the  table. 

34.  From  \  of  a  week  take  $  of  an  hour. 

(c)  To  subtract  when  the  fractions  have  a  common 
numerator : 

Multiply  the  difference  of  the  denominators  by  either  numerator, 
and  write  the  product  over  the  product  of  the  denominators. 

35.  From  £  take  f. 

1  1_8  —  5__  3         4  4_4X  3_12_  3 

5  8~~5  X8~40;  *'*  5  8~~      40     ~40~"l0'     nS* 

36.  From  $  take  yV  Ans.  ?f 

37.  From  £  take  $. 

(d)  To  take  a  mixed  number  from  a  whole  or  mixedv 
number. 

38.  From  Ci  take 

6$  —  2$  =  4£,  Ans.     (See  Art.  152,  d). 

153.   Mode  of  subtracting  when  the  fractions  have  a  common  numerator? 


COMMON   FRACTIONS.  121 

89.  From  8^  take  2^. 

In   Kx.  38,  take  $  from  J,  and  2  units  from  G  units;  but  in 

19  we  cannot  take  ^  from  -ft,  .\  reduce  one  of  the  8  units 

io  \  {,  and  add  it  to  the  .,»,,  making  \  f,  and  then  take  the  -ft  from 

$  ;•.  and  the  2  unitfl  from  the  remaining  7  units. 
1»>.  From  9£  take  3|. 

9$  —  3$  =  Si  —  3|  =  8|§  —  3^  =  Hh  Ans. 

41.  From  12$  take  4g. 

42.  From  9  take  5?.  Ans.  8ft 

43.  From  8  take  2§. 

Miscellaneous  Examples  in  Fractions. 

1.  Multiply  j3r  by  5.  Ans.  \%. 

2.  Multiply  ft  by  6. 

3.  Reduce  £§  to  its  lowest  terms. 

4.  Add  8fV  to  6H- 

-  il.tract  I8)|  from  t5tf 
G.  Reduce  23 §  to  an  improper  fraction. 

7.  Reduce  K  to  a  fraction  wboM  denominator  is  27. 

8.  Reduce  9  to  6  fractional  forms. 

9.  Divide  £|  by  ft. 

10.  Divide  V  by  V. 

11.  Divide  W  by  f. 

12.  Reduce  |  dt'  a  day  to  hours,  minutes,  and  seconds. 

1 8.  Eteduce   !pk.  5qt.  lpt.  to  the  fraction  of  a  bushel. 
1  t.   .Multiply  ,S£by  10. 

16  Divide  9|  by  l. 
ia  Divide  ||  by  9. 
17.  Divide  18  by  $. 
1 8»    B  'A  to  a  mixed  number. 

19.  Keduee  LJ|I  to  a  whole  number. 
Multiply  ||  by  | J. 

21.   Reduce  $  of  J  of  if  to  a  simple  fraction, 
in  g. 

133.   Mode  of  taking  a  mixed  number  from  a  whole  or  mixed  number  1 
11 


122  •    FRACTIONS. 

'<.  Reduce  fo  *>  and  $  to  equivalent  fractions  thai  haw 
common  denominator. 

S  I.  Reduce  \%,  ££,  and  $J  to  equivalent  fractions  having  the 
least  common  denominator. 

25.  Reduce  ^  and  §  to  equivalent  fractions  having  a  common 
numerator. 

26.  Reduce  $,  -ft,  and  $$  to  equivalent  fractions  having  the 
least  common  numerator. 

4| 

27.  Reduce  —  to  a  simple  fraction. 

28.  Add  &  to  ft. 

29.  Divide  ft  by  4. 

30.  K<  (luce  §  of  a  gallon  to  the  fraction  of  a  quart 

31.  Reduce  $  of  an  hour  t<>  the  fraction  of  a  ireek. 

32.  Reduce  f-°F  jArA°  r  v  t0  its  snnpkd  6nn- 

f  of  3  of  A  of  i7* 

33.  Multiply  §£  by  33. 
31.  Multiply  25  by  $. 
35.  Multiply  25  by  jj. 
86.  Divide  /■,  by  |. 

37.  Add  §£,  is.,  and  $d.  together. 

38.  Subtract  $  of  a  gill  from  $  of  a  gallon. 

39.  Add  fV,  VV,  fV  ^3,  and  ^  together. 

40.  From  §§  take  §g. 

•11.  Five  gallons,  3  quarts,  1  pint,  and  8  gills,  are  what  part 
of  1  gallon  ?     (See  Art.  151,  Note  3). 

42.  Three  pecks  are  what  part  of  3  pecks? 

Examples  in  Analysis. 

1»*1.  We  analyze  an  example  when  we  proceed  with 
it,  step  by  step,  according  to  its  own  conditions,  without 
being  guided  by  any  particular  rule. 

Ex.  1.   If  4  tons  of  hay  cost  $48,  what  will  7  tons  cost  ? 
Solution.    If  4  tons  cost  Sis,  then  1  ton  will  cost  \  of  $48, 
which  is  $1S  :  and  if  1  ton  cost  $12,  then  7  tons  will  00 

times  $1*2,  which  is  $84,  An-. 


COMM<>\  123 

2.   What  k  llM  value  of  12  Mm  of  land,  if  8  MTM  cost  $81? 

Ans.  $324. 
What  i>   the  cost  of  1G  barrels  of  flour,  if  3  barrels  cost 

I.  If  a  man  can  cut  8  cords  of  wood  in  4  days,  how  much 
will  M  cut  in  7  days? 

5.   If  1  ton  of  hay  costs  $15,  what  will  1,  of  a  ton  cost? 

Solution.  One  ton  costs  $15  ;  /.  J  of  a  ton  costs  £  of  $15 
=  $3,  and  $  cost  4  times  $3  =  $12,  Att. 

f>.  What  is  the  value  of  $  of  an  acre  of  land,  at  $40  per 
MN  ?  Ans.  $35. 

7.  If  f»  men  mow  12  acres  of  grass  in  a  dav,  how  many  acres 
will  they  mow  in  £  of  a  day  ? 

8.  II"  a  man  cradle  18  acres  of  wheat  in  9  days,  how  many 

..ill  he  cradle  in  ~>  d 

9.  Paid  $6  for  J  of  a  yard  of  velvet ;  what  was  the  price 
■ird? 

Solution.  Since  $6  were  paid  for  £  of  a  yard,  \  cost  £  of 
$6  =  $2,  and  .*.  |,  or  a  whole  yard,  cost  4  times  $2  =  $8,  Ans. 

10.  If  I  of  a  yard  of  ribbon  cost  G3  cents,  what  will  a  yard 
co«t  ? 

11.  If  §  of  an  acre  of  land  cost  $75,  what  is  the  price  per 
acre  ? 

1 2.  If  234  bushels  of  potatoes  grow  on  $  of  an  acre,  how 
many  busbelfl  will  grow  on  an  am? 

13.  If  J  of  a  farm  cost  $4300,  what  cost  $  of  it? 

rnoN.  If  J  cost  $1200,  Iheli  }  costs  $  of  $4200  = 
$1400,  and  |  cost  4  times  $1400  =  $5600.  Now  the  whole 
farm  100,  .-.  \  of  it  costs  \  of  $5600  =  $800,  and  $  cost 

b  times  $800  =  $4000,  Ans. 

II.  If  §  of  a  cord  of  wood  are  bought  for  $3£,  what  will  J  of 

!•">.    If   ,;,  of  |  ihSp  aie  W0H  .  what   is  the  value  of  \ 

of  h 

16.  If  \  of  the  distance  from  A  to  B  is  32  miles,  what  is  ft 
of  the  fjjatance  from  A  to  B? 


124  common  nucnoHS. 

17.  If  3  men  build  #  of  a  rod  of  wall  in  an  hour,  how  many 
rods  will  4  men  build  in  G  hours? 

18.  If  6  men  can  do  a  piece  of  work  in  3^  days,  how  long 
will  it  take  4  men  to  do  the  same  work  ? 

19.  What  cost  61b.  of  sugar,  at  8$c.  per  lb.  ? 

20.  What  shall  I  pay  for  16Mb,  of  rice,  at  4c.  per  lb.? 

21.  Bought  41b.  of  raisins,  at  12£c.  per  lb.,  and  paid  for  them 
in  Bggft,  at  16§c.  per  dozen  ;  how  many  dozen  did  it  take? 

98.  What  cost  12£lb.  of  pork,  at  6c.  per  pound? 

23.  If  f  of  a  bushel  of  wheat  co>t  $1},  what  is  the  cost  of 
12^  bushels? 

24.  If  7bl)l.  of  flour  (  what  will  3.U)bl.  cost? 

25.  If  2 1  cords  of  wood  will  pay  for  27  gallons  of  molasses, 
how  many  cords  will  pay  for  4  timet  -"  gallons  ? 

Ans.  4  times  2^  cords,  viz.  9  cords. 

26.  What  cost  12)  yards  of  silk  at  si  |  per  yar.1  ? 

27.  How  many  times  will  a  wheel  that  is  9  feet  in  circumfer- 
ence turn  round  in  running  20.\  miles? 

28.  How  many  cubic  feet  in  a  box  that  is  6^ ft.  long,  5£ft. 
wid  •,  and  3§ft.  deep  ?  Ans.  117.     (See  Art.  104). 

29.  How  many  bottles  containing  1^  pints  each  are  requited 
to  bottle  21  gallons  of  wine? 

30.  What  costs  a  farm  of  7o\  acres  at  $96^  per  acre  ? 

31.  If  it  costs  $8|  to  carry  13cwt.  3qr.  o|  lb.  8*  miles,  how 
far  Ban  the  same  be  carried  for  $16^? 

32.  Bought  ^  of  a  20-acre  lot,  and  sold  J  of  the  part  pur- 
chased ;  how  mueli  had  I  remaining? 

33.  If  3 ^  bushels  of  oats  will  sow  an  acre,  how  many  bushels 
will  sow  7£  acres  ? 

34.  A  staff  3ft.  long  east  a  shadow  £  of  a  foot  at  12  o'clock ; 
what  is  the  length  of  a  shadow  cast  by  a  steeple  125ift.  high, 
at  the  same  time? 

35.  If  a  staff  3ft.  long  casts  a  shadow  of  £  of  a  foot  at  12 
o'clock,  what  is  the  bight  of  a  steeple  that  casts  a  shadow  31|ft., 
at  the  same  time  ? 

36.  Sold  a  watch  for  $43?,  which  was  I  of  its  cost;  what  was 
its  c<> 


1 25 

.  many  pounds  of  butter  in  24  firkins  containing  33.J,  lb. 
Hid  what  IS  it  worth  at   }  of  a  dollar  per  pound  ? 
38.  If  6  mfl  number,  what  is  5}  times  that  number? 

Am.  44. 

\\  then  ',  is  |  of  •*.,  which  is  2,  and  |  are 

4  times  2  =  8.     Since  s  i>  the  number,  5}  timet  the  number 

will  be  'VI  timea  6  —  l  i,  Ana* 

If  12  is  $  of  some  number,  what  i<  7|  times  that  number ': 
40.  Fifteen  is  §  of  how  many  times  10?  .1. 

Analysis.     If  15   i-   |,  then    !;  is  ^  of  15  =  5,  and  |  are  8 
£mei  5  =  40.     Now  40  is   1  times  10  ;  .*.  15  is  §  of  four  times 

1".  A 

11.  Twenty-four  is  -ft  of  how  many  times  2?         Ans.   22. 

Thirty-five  is  |  of  how  many  times  5  ? 
43.  Seven  ninths  of  72  are  £  of  how  many  times  7  ? 

Ans.  10. 

Analysis.     Oi.v.  :.:nth  of  72  is  8,  and  #  are  7  times  8  =  5G ; 

La  |,  thru  I  \<  \  of  56,  which  is  14,  and  £  are  5  times  14  = 

70.     Now  70  is  10  times  7 ;  .-.  $  of  72  are  §  of  *e/t  times  7,  Ans. 

11.  Three  eighths  of  40  are  $  of  how  many  times  5  ? 

Ans.  7. 
45.  Seven  eighths  of  48  are  f  J  of  how  many  times  8? 
40.  Six  fifths  of  30  are  |  of  how  many  sixths  of  24  ? 

Ans.  8. 
Analysis.     One  fifth  of  30  is  6,  and  §  are  G  times  6  =  36  ; 
if  36  is  g,  then   \  is  £  of  36  =  4,  and  |  are  8  times  4  =  32. 
of  2  1  El   1.  and  4  is  contained  8  times  in  32  ;  .-.  §  of  30 
are  |  of  eight  sixths  of  2  I,  An*. 

eighths  of  M  are  $  of  how  many  thirds  of  \ 
48.   Four  sevenths  of  86  are  yC)  of  how  many  eighths  of    1"  '< 
Of  the  inhabitants  of  a  certain  town,  J  are  farmer-.  |  me- 
chanics jVj  manufacturer-.  \   students  and  professional  men,  and 
the  remainder,  cambering  246,  are  engaged  in  various  occupa- 
tion^.     What  i-  the  population  of  the  town?  An- 

What  would  be  the  population  of  the  town  mentioned  in 
Ex.  49,  all  the  conditions  remaining  the  same  except  that  240 
shall  1  Ans.  1 C  4«>. 

11* 


126  COMMON   FRACTIo 

51.  A  certain  room  is  16£ft.  long,  15ft.  wide,  and  9ft.  high; 
how  many  square  feet  in  the  walls  ? 

Ana,  T>G7.     (See  Art  101). 

52.  What  would  be  the  cost  of  carpeting  the  room  mentioned 
in  ESx.  51,  the  carpet  being  lyd.  wide,  and  costing  $1£  per  yd.? 

53.  A  merchant  bought  48^1b.  of  butter  of  one  customer,  28$ 
of  another,  25^  of  another,  and  oGfa  of  another ;  how  many 
pounds  did  he  buy,  and  what  was  the  cost  of  the  whole  at  I5& 
per  pound? 

54.  In  a  certain  school  ^  the  scholars  study  arithmetic,  £ 
algebra,  fa  geometry,  and  tin*  remainder  of  the  lehool,  viz.  14 
scholars,  study  surveying;  how  many  scholars  are  there  in  the 
school  ?  Ans.  84. 

55.  How  many  scholars  would  there  be  in  the  school  men- 
tioned in  Ex.  6  1,  if  only  seven  scholars  studied  surveying? 

5G.  A  fox  has  1 G  rods  the  start  of  a  hound,  but  the  hound 
runs  22  rods  while  the  fox  runs  20  ;  how  many  rods  will  the  fox 
run  before  the  hound  overtakes  him? 

57.  A  fox  has  18  rods  the  start  of  a  hound,  but  the  hound 
runs  25  rodfl  while  the  fox  runs  22 ;  how  far  must  the  hound  run 
to  overtake  the  fox  ? 

58.  A  boy  being  asked  how  many  doves  he  had,  replied  that 
if  he  had  as  many  more,  |  as  many  more  and  G  doves,  he  should 
have  5G;  how  many  doves  had  he? 

59.  A  boy  being  Mked  Imw  many  lambs  he  had,  replied  that 
if  he  had  twice  a^  many  more,  |  ftfl  many  more  and  5£  lambs,  he 
should  have  30  ;  how  many  lambs  had  lie  ? 

GO.  If  2  be  added  to  each  term  of  the  fraction  j,  will  the 
value  of  the  fraction  be  increased  or  diminished  ? 

Ans.  Increased  by  fa. 

61.  If  2  be  added  to  each  term  of  the  fraction  $,  will  its 
value  be  increased  or  diminished  ?        Ans.  Diminished  by  fa. 

63.  If  2  be  added  to  each  term  of  the  fraction  ■{*-,  will  its 
value  be  increased  or  diminished?  Ans.  Neither. 

What  principle  is  involved  in  the  last  three  examples?  How 
would  the  values  of  the  several  fractions  in  the  last  three  exam* 
pies  be  affected  if  2  were  subtracted  from  each  term  ? 


127 

A  merchant  owning  §  of  a  ship,  sold  $  of  his  shan  for 
$3000;  what  was  the  value  of  the  ship?  Ans.  $12000. 

54.  A  «an  tlo  a  piece  of  work  in  6  days, and  11  in  12  days;  in 
what  time  can  A  and  B  together  do  the  work? 

A.  B,  an.l  ('  can  do  a  piece  of  work  in  1  day-;  A  and  B 
lo  it  in  5  days;   in  what  time  can  ('  do  it? 

Bought  a  pair  of  oxen  and  a  hone  for  $180.    The  oxen 
of  the  price  of  the  hone  ;  what  was  the  price  of  the  l.< 

Bough!  a  pair  of  oxen  and  a  hone  for  S17o,  and  a  n 

of  the  price  of  the  bone.    The  bora  ai  much  as 

Leo  ;   what  was  the  price  of  the  wagon?  Ans.  $45. 

Six  men  are  to  be  clothed  with  cloth  thai   is   Hyd.  wide. 
Now    if  it    takes    2§y<L  of  this   cloth    for  each   man,   how   many 

yards  of  doth  |yd.  wide  will  be  sufficient  to  line  all  the  gar- 
ment 

69.  A  gentleman  gave  A-  of  his  estate  to  his  wife,  %  of  the 
remainder  to  his  BOO,  and  \  of  what  then  remained  to  his  daugh- 
ter, who  received  $37G£  ;   what  was  the  value  of  the  681 

70.  Sold  a  watch  :  which  was  $  of  its  co.-t  ;  what  was 
lost  by  the  transactk 

71.  -It*  a  man  earn  $1]  per  day,  in  how  many  days  will  he 
earn  $100? 

72.  How  many  miles  of  furrow  will  he  turned  in  plowing  an 
■ere,  if  the  furrows  are  J  of  a  foot  wide  ? 

75.   If  a  man  can  do  a  piece  of  work  in  '.)  days  by  working 
boon  per  day,  in  how  many  days,  of  8^  hours  each,  can  he 
do  the  same  work  ? 

7  1.  How  many  pounds  of  butter,  at  \  of  a  dollar  per  pound, 
will  pay  for  'J  pounds  of  i  of  a  dollar  per  pound  ? 

7"-.    It    1",    yards  of  cloth  are  required  for  1   coat,  how  many 
ta  may  It  made  from  16J  yards? 

Ik  make  a  dross,  and  3  <1:  made 

from  a  piece  containing  50  yards,  what  remnant  will  be  left? 

77.  How  many  square  feei  of  boards  will  be  required  to  make 
3  do/-'  whose  inner  dimensions  shall  be  2|  feet  in  length 

uml  breadth,  and    1 4(    feel  ED  depth,   the    hoard-   being    1    inch    in 
thickness?  Ans.  1  1 1 


128  i-Ual  fractI" 

78.  How  many  feet  will  be  required  to  make  3G  boxes  whose 
outer  dimensions  are  the  same  as  the  inner  dimensions  given  in 

Ex.  77,  the  boards  being  of  the  same  thick  td  what  is  the 

difference   in   the   capacity  of   the   two  sets  of  boxes  in  eubic 
inches.  Aus.  lOOlit. ;  lHUlc.in. 


DECIMAL     PR  ACT  [ONS. 

I*?*?.  A  Decimal  Fraction  is  a  fraction  whose  denomi- 
nator is  10,  100,  1000,  or  1  with  one  or  more  ciphers  annexed. 

Note  1.  The  word  decimal  is  derived  from  the  Latin  decern,  which  signi- 
fies ten. 

Note  2.     By  the  word  decimal  we  usually  mean  a  decimal  fraction. 

1«5G.  The  denominator  of  a  Common  Fraction  may  be  any 
number  whatever.  Every  principle  and  every  operation  in  Com- 
mon Fractions  is  equally  applicable  to  Decimal-. 

WT«  The  denominator  of  a  decimal  fraction  is  not  usually 
expressed,  since  it  can  be  .  termined,  it  being  1  with  as 

many  ciphers  annexed  as  then  are  fgures  in  the  given  decimal. 

1«58.  A  decimal  fraction  is  distinguished  from  a  whole  num- 
ber by  a  period,  called  the  decimal  j»>i/it  or  sepuratri.r.  pi 
before  the  decimal;  the  first  figure  at  the  right  of  the  point  is 
(ruths;  the  second,  hundredths;  the  third,  thousandths;  etc.; 
thus,  .6  =  ^,  .06  =  T$s,  .0QG  =  TJfoj5y  etc.,  the  figures  in  the 
decimal  decreasing  in  value  from  left  to  right,  as  in  whole  num- 
Art.  15). 

l",r».  What  is  a  Decimal  Fraction?  Decimal,  from  what  derived?  What 
i-<  usually  meant  by  the  word  decimal?  156.  A  Common  Fraction,  what  is  its 
denominator?  Are  the  principles  of  common  fractions  applicable  to  decimals? 
157.  Is  the  denominator  of  a  decimal  usually  expressed?  158.  How  is  a 
decimal  traction  distinguished  from  a  whole  number?  What  is  the  first  figure 
at  the  right  of  the  point?    Secoud?   Third? 


!•?!).    since    whole   numberi    ind    decimal   fraction!   both 

aae  by  tlu'  Mime  law  fVdii)  lrit  to  right,  they  may  be 

r  in  the  same  example,  and  numerated  a-s  in  the 

following 

NUMERATION  TABLE. 


9 

a 

«r  .  Hi? 


w        r—  3  I  '•—  •  C»  2  »  —IS  ■ 

"=  :    -r    ~    =    ~    —    S^-tSu 


i?    /  3  *l       *?  'S  •§  ^  •§ 


c     s     -     fi    —     A     c 


8    4    7    1.    8    8    8    8    7    2    8    4    3    2    1    6 


1GO.  A  whole  number  and  decimal  fraction  written  together, 
m  in  the  aboTe  table,  form  a  mixed  number.    The  integral  part 

is  numerated  from   the  decimal   point  toward  the  left,  and   the 
fraction  from  the  same  point  toward  the  tight,  each  figure,  both 

in   the  whole   Dumber  and  decimal,  taking  i/s  inline  and  value  by 

its  distance  from  the.  decimal  point,     Hence, 

101.  Moving  the  decimal  point  one  place  toward  the  right, 
multipb'ei  the  number  by  1<>;  moving  the  point  two  places  mul- 
tiplies   the    Damber    l»y    100,  etc,      AJfO    moving    the    point    one 

to  the  leftjdwidet  the  number  by  lOj  moving  the  [joint  two 
divides  by  LOO,  etc. 
1<»£.  la  reading  a  d.cinial.  we  may  give  the  name  to  each 
figure  separately,  or  we  may  read  it  as  we  read  a  whole  Dumber, 
and  (jirc  tin-  name  of  the  right-hand  figure  only;  thus,  the  expres- 
sion  £8  may  be  r.ad  {-0  and  r§y,  or  it  may  he  read  ffc,  for  ^ 
and  , 

150.  "^  imcnitiuii   liitilo.     100.    What  h  a  mixed  Dumber?     Wliic-h 

wejr  to  the  Integral  part  numerated  1    VTbJeb  way  tbe  decimal?    What 
M  of  a  Iftfre  I    ir, i.    i  orina;  tbe  d< 

i<>i»it  :.»  t'i<-  right  eflbel  f  a  namber?     How  mortog  it  to  tbe  i<fi* 

I  wliat  two  v.  .iv 


130  DECIMAL   FRACTK 

163.  To  read  a  decimal  fraction  as  we  read  a  whole  num- 
ber, requires  two  numerations  ;  first,  from  the  decimal  point,  to 
determine  the  denominator!  and  second,  towards  the  point,  to  de- 
termine the  numerator  ;  thus,  to  read  the  following:  .3578G92, 
first,  to  determine  the  denominator  or  name  of  the  right-hand 
figure,  beginning  at  the  3,  say,  tenths,  hundredths,  thousandths, 
ten-thousandths,  hondreoVthonsandths,  millionths,  ten-millionths ; 
and  then,  to  determine  the  value  of  the  numerator  y  or  name  of 
the  UfiJtand  figure  considered  (ts  an.  inUgerx  beginning  at  the 

\ ,  units,  tens,  hundreds,  then  sands,  tens  of  thousands,  hun- 
dreds of  thousand-,  millions,  and  then  read,  three  million,  fi\c 
hundred  and  seventy-eight  thousand,  six  hundred  and  ninety- 
two  te)i-)iullioutliS. 

164.  Since  multiplying  both  terms  of  a  fraction  by  the  lame 
number  does  not  alter  ill  value  (Art.  1 17.  a.  Note  1).  annexing 
one  or  more  ciphers  to  a  decimal  does  not  affect  iti  value;  thus, 
A  =  flfr  =  r  ■  !-  *  -2  =  -20  =  -200- 

16*1.  Prefixing  a  cipher  to  a  decimal,  i.  e,  inserting  a  cipher 
between  the  leparatrU  and  a  deeima]  figure,  aVatftntsae*  the  value 
of  that  figure  to  -fa  its  previous  value;  for  it  removes  the  fig- 
ure one  place  further  from  the  decimal  point  (Art.  1G1);  thus, 
.3  =  fa  but  .03  =  only  !  ;\(,  which  ii  but  fa  of  fo 

What  is  the  effect  of  prefixing  two,  three,  or  more  ciphers  to 
a  decimal  ? 

100.     A  common  fraction  is  sometimes  annexed  to  a  decimal  ; 

21 
thus,  ,2J.     This  is  equivalent  to  the  complex  fraction  -  .     The 

common  fraction  is  never  to  be  counted  as  a  decimal  place,  but 
it  is  always  a  fraction  of  a  unit  of  that  order  represented  by 
the  preceding  decimal  figure;  thus,  in  .234^,  the  |  ifl  half  of  a 

ndth. 

163.  To  read  a  decimal  requires  how  many  numerations?  Firet.  which  way! 
For  what  purpose?  Second,  which  way?  For  what?  Illustrate.  164.  How 
is  the  value  of  a  decimal  affected  hy  annexing  a  cipher?  Why?  165.  How 
by  prefixing  a  cipher?  Why?  166.  A  common  fraction  annexed  to  a  decimal, 
what  is  it?     1  !1u*trate. 


DECIMAL 

M,     \l    m  I   'i   I    IM  M.      J'l:A«     !  tO» 

107.    Lei  the  pupil  the  following  nunv 

l.  Fifty-two  hundredths. 

I,  Pour  hundred  md  sixteen  thousandths.  Am.  .11 G. 

brae  hundred  end  forty-two  tan-thontandthw, 

: .  1 .  An  ambiguity  often  arises  in  enunciating  1  whole  number  and 
u  decimal  in  the  same  example  ;  thus,  JQ3  is  two  hundred  end  three  thou- 
sandths, ami  200.003  is  two  hundred,  and  time  thousandths.  This  ambi- 
guity  may,  however,  be  avoided    by  placing   the  word  di<iiu<il   before   tho 

m;  thus,  200.003  may  be  read  two  hundred  end  decuaot  three  thou- 
sandtljs. 

N .  >  1 1 .  _'  In  decimals,  as  in  whole  numbers  (Art.  1C),  ciphers  arc  used 
to  till  placet  that  would  otherwise  be  vacant. 

4.   Write  the  decimal  >ix  hundred  and  forty-one  thousandths. 
Decimal  five  hundred  and  eighteen  teu-thou.-andihs. 

•'..  Bight  hundred  ami  decimal  eight  thousandths. 

7.  Six  thousand  and  decimal  six  miliionths. 

8.  Nine  hundred  and  thirty  and  eight  tenths. 

'.'.    Decimal  two  hundred  and  forty->ix  ten-millionth*. 
1<>.   One  thousand  and  decimal  two  hundred-thou>nndths. 

11.  Eleven  and  eleven  ten-billiontbs. 

12.  Six  hundred  ami  sixteen  and  sixteen  trillionths. 

13.  Ten  thousand  and  decimal  four  ten-thousandths. 

I  1.  Decimal  three  hundred  twenty-live  thousand,  four  bun- 
dred  and  eighty-seven  hundred-millionth-. 

1GS.  Write  the  following  numbers  in  words,  or  read  them 
orally  : 

7.  8694.876942 

8.  760.4071 

9.  4004.40040004 

10.  839  ;:;;; 

11.  46,00046482 

12.  8769.27642! 

What  uncertain-  -:«    in    nailing   iiu'm  «I  iiiiihImt- '      ll'»w  cau 

thin    ainl.i-uity    be   a\  oiilcl '      lor    what   are   ciphers   u.*ed    iu    tin 


1. 

i2A 

■2. 

::.789 

1. 

10045 

6. 

1.8< 

132  DECIMAL    TRACTIONS. 

Note  1.  Addition,  subtraction,  multiplication,  and  division  of  decimal 
fractions  are  performed  precisely  ii  the  ihjm  operation!  in  whole  nam 

no  further  explanation  l»eii  ,  except  to  determine  the  place  of  the 

decimal  point  in  the  several  results. 

Note  2.     The  proofs  arc  the  same  as  in  whole  num!>ers. 

Problem  1. 
109.    To  add  decimal  fractions: 

Rule.  Place  tenths  under  tenths  hundredth*  under  hun- 
dredths, etc.;  tJien  add  as  in  whole  numbers,  and  place  the  point 
in  the  sum  dinctlij  under  the  points  in  the  number*  added. 


Ex.  1. 
3  6.4  7  8 
8  4.9  2  G 
2  8.0  4  7 

2. 
84  8.842 

3  8  7.6  4  G 
9  8  4.2  8  5 

ft. 

5  G  4.9  8  7  4  2  6 

1  2.8  G  5  3  9 
8  7  4.8  2  7  6  4  1 

Sum,     1  4  9.4  4  5 

1  7  2  8.7  7  3 

1  4  8  2.6  8  0  4  5  7 

Proof,    1  4  9.4  4  5 

1  7  2  8.7  7  3 

1482.680457 

4.  5. 

8  7  2  1  4  3.8  7  2  9  34820  9.1  534268741. \ 

24  10.40  2*  2  7o.i  23 

7  9  1  8  4  2.2  1  G  3  9  l  2  9.8  7  251842172 

841.3  60498  86512  3.7  1942 

7  2  4  3  1  0.0  0  6  8  4  3  31981  7.0  5841628347 

6.  Add  42.76,  934.247,  27.862.  Ans.  1004.8G9. 

7.  Add  3.54G,  44.8693,  2.8769,  and  784.68728. 

8.  872.34,  6789.3274,  22.987,  and  346.4 2. 

9.  Add  3582.47,  62.84693,  .47249,  ami  7.458. 

10.  Add  five  hundred  and  decimal  six  thousandths;  forty-five 
millionths  ;  eighty-four  million  and  decimal  twelve  millionth*  ; 
seventy  thousandths  ;  and  decimal  three  hundred  and  fifty-four 
hundred-thousandths.  Ans.  84000500.079597. 

11.  What  is  the  sum  of  one  thousand  two  hundred  twenty-six. 

168.    How  are  addition,  subtraction,   multiplication,  and  division  of  decimals 
performed?    Proofs?     1G9.    Rule  for  addition?    The  point,  where  placed? 


fi.nl  decimal  one  hundred  :m<l  forty-fair  thmtaandthn  i  twenty- 
•vu  hundredth*)  and  eight  hundred  forty-nine  and 
•three  hundredth*  ? 

12.  What  i-  tin-  sum  of  fifty  hundred-thousandths;  eighteen 
hundred  and  decimal  Bixty-three  ten-thonsandthfi ;  .-evenly -four 
and  Berenteen  hundred-thousandth 

Pboblxm  2. 

17©.   To  subtract  a  less  decimal  from  a  greater: 

ivi  i.k.     Place  the  less  number  under  the  greater^  truths  under 

.  ;  then  subtract  as  in    whole  numbers,  and  place   the 
MMndert  directly  under  the  points  in  the  min 
and  subtrahend. 


From 

.  1. 
2.8  9  4  6 

2. 

4  7.4  2  9  6  4 
18.1  62  9  3 

3. 
4  9.0  6  8  4 
2  1.8  7  4  69  3 

Bern. 

3.5  3  3  3 

2  9.2  6  6  7  1 

27.198  70  7 

Proof, 

6.4  2  7  9 

4  7.4  2  9  6  1 

i  .      If.  as  in  Ex.  9,  there  are  rnon>  figures  in  the  subtrahend  tlian  in 
the   minuend,  the  deficiency  may  be  supplied  hy  annexing  ciphers,  or  sup- 

tnem  annexed,  to  the  minuend  (Art  164). 

1.  From  65.8487  take  24.3869.  Ana,  11.1618. 

:>.  From  1684.469  take  968.8749 

6.  From  9846.2764  take  5427.9824. 

7.  From  21  W.6872  take  L724.19 

■  one  thousand  eight  hundred  seventy-six  and  deci- 
mal three  hundred  sixty-lour  thousandths,  take  eight  hundred 
D  and  decimal  three  hundred  and  three  thousandths. 

Ana.  1060.061. 

9.   From  U-n  take  six  millionth.*. 
]<».  A  man  owned  eightj-eeren  hundredths  of  a  railroad  and 
•old  forty-eight  hundredths  of  it;  what  part  of  the  road  did  be 
still  own  ? 

■  :<».     I     I  for  subtraction  of  decimals?     When  the  number  of  «lccim:i: 
in  the  subtrahend  exceeds  the  number  of  urcimal  plftQM  in  thfl  BUM 
If 


134  bECIMAL   FRACTION. 

Problem  3. 
171.    To  multiply  one  decimal  by  another: 

Rule.  Multiply  as  in  whole  numbers,  and  point  off  as  many 
figures  for  decimals  in  the  product  as  there  are  decimal  places  in 
both  factors,  counted  together. 

;    Ex.  1.  Multiply  .48  by  .26. 

OPERATIOX.  PROOF. 

Multiplicand,  .4  8 

Multiplier,  .2  6  .48 

288  208 

96  104 

Product,  .1248         =         .12  48 

(a)  If  the  number  of  figures  in  the  product  is  less  than  tho 
number  of  decimal  places  in  the  two  factor-,  the  deficiency  must 
be  supplied  by  prefixing  cipher*  to  the  product,  as  in  Ex.  3. 

2.  3. 

Multiplicand,       2  6.2983  .::  2 

Multiplier,        8^  .2  3 

1051932  ~9~6 

2  I  03864  64 

Product,        2  2  0.9  0  5  7  2  .0736 

Note  1.    The  reason  of  the  rule  for  pointing  the  product  will  Ik)  ol>\ 
if  wt  change  die  decimals  to  the  form  of  eotnmon  fraction!  and  then  per* 
form  the  multiplication  ; 

Thus,    .48  x  .2*6  =  iW,  X  A«j  =  flflftfr  =  .1248,  as  in  Ex.  I. 

Again,  .32  X  .23  =  ^  X  A\  =  i lllu  =  -°"36,  as  in  Ex.  :). 

N«»tk  2.  The  reason  of  the  rule  for  pointing  the  product  may  be  ex- 
plained in  another  maimer,  as  follows  : 

The  smaller  the  factors  arc,  the  smaller  is  the  product.     Now,  by  trial,  wo 
know  that 
32X2  3=      73G;  .-.,  dividing  one  factor  by  10  (Art.  1G1 ),  we  have 
•  J  2X  2.3  =      7  3.6  =  A  °f tnc  previous  product ;  dividing  again  by  10, 
3  2  X  .2  3  =      7.3  6  =  tV  of  the  2d  product ;  dividing  the  other  factor  by  1 0, 
3.2  X. 2  3  =     .736  =  j\y  of  the  3d  product ;  dividing  again  by  10, 
.3  2  X  .2  3  =  .0  7  3  6  =  tV  of  the  4th  product ;  dividing  again  by  10, 
.03 2 X  23  =.00 736=  A  of  the  5th  product;  and  so  on  to  any  extent 


.3  2  5  9 

.00  00  2  5 

i  e  a 

6518 

.0  000081475 

Ans.  .208754. 

Ans.  156.915. 

Ans.  182 

Aik  .0082824. 

Ans.  .0000204. 

4. 
Multiplicand, 

Multiplier,  .5  l 

1  6944 
21180 

Product,       2  2  8.7  4  4 

&  Multiply  .5642  by  .37. 

7.  Multiply  34.87  by  4.5. 

8.  Multiply  2769  by  .84. 
Multiply  .2436  by  .034. 

10.  Multiply  .0068  by  .003. 

11.  Multiply  86.874  by  .5421. 

12.  Multiply  .14687  by  .00054. 
IS.  Multiply  .17288  by  .11403. 
11.  Multiply  .00869  by  .24683. 

15.  Multiply  8.756  by  10.  Ans.  87.56  (See  Art.  161). 

16.  Multiply  356.4  by  100.  Ans.  35640. 

17.  Multiply  9.8765  by  1000. 

18.  Multiply  348.09  by  100000. 

19.  Multiply  286.487  by  100000. 

20.  Multiply  37438  by  100000. 

21.  Multipfy  4.68  by  20.  Ans.  93.6. 
In    Ex.  '21  multiply  by  the  factors  of  20,  viz.  10  and  2;  i.  c 

move  the  point  oik-  place  to  the  right,  and  then  multiply  by  -,. 

22.  Multiply  86.42  by  60.  Ans.  2185.2. 
28.  Multiply  472.8  by  800.  An-.  878240. 
24  Multiply  86.74  by  300. 

Multiply  54.26  by  406000. 
26.  Multiply  throe  hundred  and  fifty-six  thousandths  by  one 
hundred  and  forty-five  ten-thoii>andth>.  An-.  .<mi.",1G2. 

Multiply  thirty-four  milliontlis  l>y  t\venty->ix  ten-millionth-. 
Multiply  eight  hundred  and  forty-two  thousandths  l.\ 

hundred  thousand. 

171.     Kuie    for    mttfttpllOStton    of  decimals'      -  re    are  not    tiguns 

b  in  t he  product?     PWMOM  Of  t Iio  rule  for  pointing  the  prod;. 
explnn:e 


136  DECIMAL  FRACTI" 

Problem  4. 
173.    To  divide  one  decimal  fraction  by  another : 

Rule.  Divide  as  in  whole  numbers,  and  point  off  as  many 
figures  for  decimals  in  the  quotient  as  the  number  of  decimal 
places  in  the  dividend  exceeds  those  in  the  divisor. 

Ex.  1.  Divide  .625  by  .25. 

OPERATION.  rilOOF. 

.2  5  ).G  2  5  (2JS  £6  Divisor. 

5  0  2.5  Quotient 


i %  6  125 

125  50 


0  2  5  Dividend. 

2.  Divide  1.257*  Ans.  .508005. 

3.  Divide  8.4364*  .06.  Ans.  140.608018. 

(u)  If  the  number  of  figWQf  10  the  QOOtlont  i.>  leu  tlian  the 
excess  of  decimal  places  in  the  dividend  over  those  of  the  divi- 
sor, supply  the  deficiency  by  prefixing  ciphers  to  the  quotient, 

4.  Divide  .000744  by  .62.  Ans.  .0012. 

Notk  I,     The  dividend  is  a  product,  the  divisor  and  quotient  being  the 
.'t.  77)  ;   hence  the  rule  tbt  pointing  the  quotient. 

Note  2.  The  rule  for  determining  the  place  of  the  point  in  the  quotient 
may  alio  he  explained  by  changing  the  decimals  to  the  form  of  common 
fraction*  and  performing  the  Hviaiea ;  thus, 

-•25  =  T^V-i2A  =  ?3  =  2.5. 

m  3.  By  attending  to  the  relative  size  of  divisor  end  dividend  (Art. 
83),  we  have  another  mode  of  fixing  the  place  of  the  decimal  point  in  *.he 
quotient ;  thus, 

G  2  5  -f-  2  5  =    2  5  ;  .*. ,  by  dividing  the  divhliml  by  10  (Art.  161),  we  have 
6  2.5  -f-  2  5=    2.5  =  tV  of  the  preceding  quotient ;  dividing  again  by  10, 
6.2  5  -I-  2  5  =  .2  5  =  yV  of  the  2d  quotient ;  dividing  again  by  10, 

r  25  =.0  2  5  =  iV  of  the  3d  quotient.    Now  dividing  the  divisor  by  1<>, 
,6  8  5    r  2.5  =  .2  5  =  10  times  the  4th  quotient ;  dividing  again  by  10, 
.6  2  5  -r  .2  5  =    2.5  =  10  times  the  5th  quotient;  and  so  on  to  any  extent. 

17  4.  Rate  for  dividing  decimals?  "What  is  said  of  ciphers  la  the  quotient? 
Reason  of  the  rule  for  pointing  the  quotient?    Second  explanation?    Third? 


DECIMAL    IKAtTI 

Divide  88.742$  bj  .  Ant,  L6497. 

Aas.  .30. 
7.   Divide  .000975  by  .15. 
to  17.472  by  .48. 

1.77  11  bj   '■■-Ml. 
ia  Divide  08.794  by  1l\:;. 

.  (h)  If  there  tie  more  decimal  places  fa  the  divisor  than  in 
the  dividend,  the  number  may  be  made  equal  by  annexing  one 
or  more  ciphers  to  the  dividend.  The  quotient  will  then  be  a 
whole  number ;  thus,  L6  -~  .18  =  4.50  -*-  .18  ==  25. 

11.  Divide  8647  by  .125.  Ans.  2917G. 

12.  Divide  90821.6  by  8.642.  Ans.  24800. 
1.;.   Divide  72  by  .064. 

(c)  If  there  fa  a  remainder  after  all  the  figures  of  the  divi- 
dend have  been  used,  the  division  may  be  continued  by  annex- 
iphera  to  the  dividend.     Bach  cipher  annexed  bacon 

decimal  place  in  the  dividend. 

In  some  examples  this  operation  may  be  continued  until  there 
i-  no  remainder,  but  in  others  there  will  necessarily  be  8  remain- 
der, however  far  the  operation  may  be  continued.  This  latter 
dam  of  examples  give-  rise  to  circulating  decimals ;  thus,  .7  — 
.9  =  .7777,  etc.     Again,  .8  ~  .11  =.727272,  etc    In  the  first 

of  these  example-,  the  figure  7  will  be  repeated  perpetually,  and 
in  the  Second  example,  the  figures  7  ami  2  will  be  repeated  in 
like    manner.      Whenever   the   remainder  consists  of   the    -aim- 

figure  or  figures  a-  any  preceding  dividend,  the  quotient  figures 
will  begin  to  repeat 

It  may  be   remarked,  however,  that,  if  the  divisor  contains  no 

prime  factors  but  2'fl  and  .Vs,  the  divison  can  <dwai/s  be  continued 

until  there   .shall    be    no   remainder;    but  if  there   is   any  other 

prime   fa. -tor   in  the  divisor,  the  division  can  nrver  be  completed 

the  same  oth-  fa  in   the  original  dividend;  for  a 

17  i.  w  hut  shall  bo  done  when  there  are  more  decimal  places  in  the  divisor 
than   in  the  dividend  .'     What    is   done  when    there   is  |  reinaindei  ' 

•I  is  what.'     When   can  the   dlt  tnjiletcd.'     When  can  it  not  b« 

completed'    Win' 

12* 


138  DECIMAL  FRACTIONS. 

dividend  is  not  divisible  by  a  divisor  unless  it  contains  all  the 
factors  of  the  divisor ;  whereas  annexing  ciphers  to  the  dividend 
introduces  no  prime  factor  into  it  except  2's  and  5's. 

14.  Divide  .13  by  8. 

15.  Divide  7.2  by  .16. 
1G.  Divide  8.7  by  .25. 

17.  Divide  3.G  by  7.5. 

Note  4.     "When  a  decimal  is  not  complete,  wc  sometimes  place  tho  K^n 
4-  after  it,  signifying  that  there  is  a  remainder. 

18.  Divide  .34  by  J  1.  Ass.  1.41G6+. 

19.  Divide  .73  by  1.5. 

20.  Divide  4.68  bj  2.9. 

21.  Divide  3G.5  by  10.  Ans.  3.65  (See  Art.  1G1). 

22.  Divide  4.69  by  100.  Ans.  .0469. 

23.  Divide  816.9  by  100. 

24.  Divide  5.647  by  1000. 

25.  Divide  843.57  by  300.  Ans.  2.8119. 

In  Ex.  25,  divide  by  the  factors  of  300,  viz.  100  and  3 ;  i.e. 
move  the  point  two  places  to  the  left  and  then  divide  by  3. 

26.  Divide   >.G412  by  400.  Ans.  .009103. 

27.  Divide  56.487  by  8000. 

28.  Divide  :5G.49  by  600. 

Divide  three  thousand  eight  hundred  end  fifty-three  hun- 
dred-thousandths by  thirty-two  millionth*,        An-.    1204.0625. 

30.  Divide  eighty-four  and  eighty-four  hundredths  by  forty- 
eight  thousandths. 

Problem  5. 
173.    To  reduce  a  common  fraction  to  a  decimal. 
Ex.  1.  Reduce  £  to  a  decimal  fraction. 

I  X  100  =  a$ft  =  75  ;  and  75  -f-  100  =  .75,    Ans. 

If  a  number  be  multiplied  by  any  number,  and  the  product  be 
divided  by  the  multiplier,  the  quotient  will  be  the  multiplicand 

172.    For  what  is  the  sign  -j-  sometimes  used? 


DECIMAL    PBACTIOl  139 

in  the  above  example,  |  is  multiplied  by  100 

nnexing  two  riphen  t<>  the  numerators  the  fracti 

then  reduced  to  the  whole  number  7"»,  and,  finally.  7">  ii  divided 

by   LOO  by  placing  the  decimal  point  before  the  7'>  j  .-.  $  =  .75. 

Hen 

Rl  i  r.      Annex  one  or  more  ciphers  to  the  numerator  and  di- 
vide Uie  result  by  the  denominator.  cottinuinrf  the  operation   until 
Is  no  remainder,  or  as  far  as  Is   desirable.      Point  off  us 
man;/  decimal  places  In  the  quotient  as  there  are  ciphers  am 
to  the  numerator. 

•J.   Etedoce  $  to  a  decimal  fraction. 
I  X  1000  =  a^-a  =  375  ;  and  375  h-  1000  =  .375,  Ans. 
5.   Reduce  fa  to  a  decimal.  Ans.  .4375. 

4.  Reduce-  H  to  a  decimal.  Ans.  1.140625. 

5.  Reduce  $$  to  a  decimal. 

6.  Reduce  fa  to  a  decimal.  Ans.  .5833— (-. 

7.  Reduce  £  to  a  decimal.  Ans.  .3333+. 

8.  Reduce  9  to  a  decimal.  Ans.  .428571+. 

9.  Reduce  I,  I  I  ||,  lif  ^  and  *J  to  decimals. 

171.     Every  decimal  fraction  is  a  common  fraction,  and,  if 
nominator  be  written,  it  will  appear  as  such.     It  may  then 
[need  to  lower  terms,  or  modified  like  any  other  common 
fraction*      This  proves  the  rule  in  Art.   17.'5. 

10.  Reduce  .48  to  the  form  «>f  a  common  fraction  and  then  to 
it>  lowed  term-.  .48  =  ^*0  a=tt>  Ant, 

1 1.  Reduce  .125  to  its  lowest  terms. 

•125  =  A2A  =  M  =  A  =  b  Ana, 

12.  Reduce  .17  to  the  form  of  a  common  fraction. 

Ana,  tVj. 

13.  Reduce  .27  ■  KX)25,  and  ,00 

14  J.8.  2.8  =  f§  =  V>  An<. 

15.    :  \  and  2.0 1 

17.1     I:,',,     for   rtdndBf   a   common    fraction   to   a   decimal'      Explanation* 
.•imal  al?o  a  common  fraction?      How  is  this  made  evident'     How 
iuli'  in  Ait.   173  be  proved  MM 


140  DECIMAL   FRACTIO 

Problem  C. 

175.  To  reduce  whole  numbers  of  lower  denomina- 
tions to  the  decimal  of  a  higher  denomination. 

Ex.  1.  Reduce  2pk.  ?x\t.  to  the  decimal  of  a  bushel 
1st.  3qt.  =  gpk.  =  .375pk. ;  .-.  2pk.  and  3qt.  =  2.37."ipk. 
2d.   2.375pk.=  ^Y-J'•bush•  =  •'V•,;j7:',,l,^1•■  Ans. 

The  principle  is  the  same  as  in  Art.  17.5.     Hence, 

Hull.  Hearing  annexed  one  or  more  ciphers  to  the  lowest  de- 
nomination, divide  by  the  number  it  takes  of  that  denomination  to 
make  one  of  the  nest  hiyJa r,  and  annex  the  quotient  as  a  decimal 
to  that  next  higher  ;  (hen  divide  the  result  by  the  number  it  takes 
of  THIS  denomination  to  make  one  of  the  next  higher,  and  so 
continue  till  it  is  brought  to  the  denomination  required. 

2.  Reduce  9s.  Gd.  3qr.  to  the  decimal  of  a  pound. 

on:  RATION*. 

3.0  0  qr. 


4 
12 
20 


<..7  5  OOcL  3qr.  =  .75d. ;  C.75d.  =  .5636s.  ; 


9.5  6  2  5  0  0  s.  9.5C25s.  =  .478125£,  Ans. 


.4  7  8  1  2  5  £,  Ans. 

Note.     In  dividing  by  20  to  reduce  the  decimal  of  ft  pound,  and  in  all 
similar  examples,  we  may  point  off  the  0  in  the  divisor,  and  then  divide  by 
2,  but  in  Mich  u  case  the  point  in  the  dividend  must  be  moved  one  place  t< 
the  left,  for  by  so  doing  both  divisor  and  dividend  are  divided  by  10,  and 
.-.  the  quotient  is  unchanged  (Art.  84,  hj. 

3.  Reduce  2ft.  9in.  lb.  c.  to  the  decimal  of  a  yard. 

OPERATION*. 

1.0  0  0  0  0  0  b.  c.  In     this     example 

9.3  3  3  3  3  3  -f  in.  t,"'n'   wil-   be   a  re" 


3 

12 
3 


mainder,  however  far 


.9  2  5  9  2  5  +  yd.,  Ans.  ried. 

4.  Reduce  3cwt.  2qr.  201b.  8oz.  to  the  decimal  of  a  ton. 

175.  Rule  for  reducing  the  lower  denominations  of  a  compound  number  to 
the  decimal  of  a  higher  denomination?  Principle?  Mode  of  dividing  when  the 
divisor  is  20,  40,  etc  '     When  the  divisor  is  a  mixed  number? 


QIAL   i  B  \<  ii'  1  11 

.../..  LSdwt  I8gr,  to  the  decimal  of  a  pound,  Troy 

ht.  Ans.  .303125. 

imal  of  B  pound. 
7.    Reduce    5yd,    8ft.    Sin.    to    the    d.-eimal    of   :i    rod,    Long 

ire. 

on  Since  one  of  the  di- 

1  2 

3 


2 

1  1 


J...  0  00  ft. 

5.8  3  3  3  +  yd. 
2 

11.6666  +  half 

yd. 

6.0in.  ra,  in  this  example, 

U   51,  both   divix.r  and 
dividend  air  reduced  to 

halve*.     The  feel  and 

inches  are  more  than  a 
half    yard  :    ,\   the   gum 
\  06_j_rods  "Ans>  Of  the-  given  numbers  il 

1  more  than  a  rod. 

8.  Reduce  3s.  15°  30"  to  the  decimal  of  a  circumference. 

Ana.  .291689+. 
Reduce  :M.  Oh.  18m.  fieec.  to  the  decimal  of  a  week. 

10.  Reduce  2qt.  lpt.  lgi.  to  the  decimal  of  a  gallon. 

1 1.  Reduce  War.  Boh,  Sid.  lOli.  to  the  decimal  of  a  mile. 
1  J.    Reduce  8cu.  ft.  144c.  in.  to  the  decimal  of  a  cubic  yard. 

U educe  3r.  2rd.  20yd.  to  the  decimal  of  an  acre. 
1  t.   Reduce  5fur.  30rd.  5yd.  1ft.  9in.  2  b.  c.  to  the  decimal  of  a 
mile. 

Probum  7. 

176.   To  reduce  a  decimal  of  a  higher  denomination 
to  whole  numbers  of  lower  denominations. 

Ex.  1.   Reduce  .  12.si:?o£  to  .-hillings,  pence,  and  farthings. 

OI'KKAI   ! 

£.  1  '2  s  1  This  article  is  the  reverse  of  Art. 

2  0  175;  .-.  litvt  multiply  by  20,  because 

")  2  5  0  0  s  then;  will  be  20  times  a-  many  shil- 

,  2  luuHi  aa  pounds.     Pot  alike  reason, 

multiply  the  fractional  part  of  a  shil- 

i  500d.  Kngby  12,  to  reduce  it  to  pence,  etc. 

4  After  having  fixed  the  decimal  point 

,i(,r.  in  the  several  products,  the  <i]>lt<rs  <u 

f/ir   BIGHT  of  the  significant  figure* 
Iqt.  are  disregarded. 


142  DECIMAL   FRACTI' 

Rule.  Multiply  tJte  given  decimal  by  the  number  it  takes  of 
the  next  lower  denomination  to  maJce  one  of  this  higher \  and  place 
the  decimal  point  as  in  multiplication  of  decimals  ;  multiply  the 
decimal  part  of  this  product  l>y  the  number  it  takes  of  the 
next  lower  denomination  to  make  one  of  this,  and  so  proceed  as 
fir  as  necessary.  The  several  numbers  at  Ute  left  of  the  points 
tvill  be  the  answer, 

2.  Reduce  .984375  of  a  bushel  to  pecks,  quarts,  and  pints. 

Ans.  3pk.  7qt.  lpt. 

3.  Reduce  .40625  of  a  gallon  to  quarts,  pints,  and  gill>. 

4.  Reduce  .902288  of  a  lunar  month  to  weeks,  days,  hours, 
minutes,  and  second-.  An-.  $w.  id.  C>h.  80m.  1  ">.12'J0sec 

5.  Reduce  .90625  of  a  yard  to  quarters,  nail- 

6.  What  i>  the  value  of  .37.">°  ?  Ans.  22'  30". 

7.  What  is  the  value  of  .375  of  a  ton  ? 

8.  What  is  the  value  of  .4658  of  a  pound,  Troy  Weight  ? 

9.  Reduce  .3587  of  a  mile  to  furlongs,  rods,  yards,  etc. 
10.  Reduce  .5621b  to  5,  3,  etc 

Miscellaneous  Examples  in  Decimal  Fractions. 

1.  What  is  the  cost  of  6.251b.  of  beef,  at  12  cents  per  pound? 

Ans.  7oc. 

2.  Bought  4.5  tons  of  hay,  at  $12.50  per  ton;  what  was  the 
cost  of  the  whole?  An-.  $56. 

3.  What  i-  the  raloe  of  8  acres  of  land,  at  $62.50  per  acre? 

4.  Paid  >">00  for  8  acres  of  land;  what  was  the  priee  per 
acre? 

5.  Paid  $500  for  a  piece  of  land  at  $02.50  per  acre ;  how 
many  acres  were  bought  ? 

6.  Bought  land  at  $62.50  per  acre,  and  sold  it  again  at  $75 
per  acre,  thereby  making  $100 ;  how  many  acres  were  bought? 

7.  Bought  8  acres  of  land  at  $62.50  per  acre,  and  sold  the  lot 
|600  ;  was  there  a  gain  or  a  loss?     How  much  total?    How 

much  per  acre  ? 

J  76.    Rule  for  reducing  a  decimal  of  a  higher  denomination  to  whole  num- 
bers of  lower  denominations?    Explanation? 


m:<  imal   IRA4  ii'  143 

B,  v> '!.••  Br.  20rd.  of  hud,  aft  $40  per  acre  ? 

:  St.  I5cwt  lojr.  12fclb.  of  coal,  at  $G  per  ton? 
in.  What  oosi  12.25  cords  of  woo.  1,  at  $6  per  cord  ? 

1 1.  What  cost  7}  oorda  of  wood,  at  $64(5  per  oord  ? 

12.  What  will  it  cost  to  build  24m.  Sfbr.  20rd  of  railroad,  at 
$3775  per  mile  ? 

18.  A  rectangular  field  is  40.5  radi  long,  and  30.5  rods  wide  ; 
wliat  will  it  cost  to  build  I  wall  around  it,  at  $1  per  rod? 

1  1.  What  coal  5yd.  Bqr,  2na.  of  cloth,  at  16c.  per  yard  ? 

1").  How  much  land  in  a  rectangular  field  that  is  -10.5  rods 
i  rods  wide  ? 

in.  Whal  would  16  hales  of  cotton  cost,  each  bale  weighing 
tftewt,  at  $10.50  per  cwt? 

17.  What  0081  .825  of  a  ton  of  coal,  at  S7  per  ton? 

18.  What  OOat  .825  CWt  Of  coal,  at  ^7  per  ton? 

l'.t.  What  is  the  value  of  .25  of  a  ton  of  hay,  at  2£  5s.  Gd.  lqr. 
per  ton  ? 

20.  What  is  the  value  of  .7")  cwt.  of  hay,  at  2£  5s.  Gd.  lqr. 
per  ton  ? 

21.  Taid  3£  9s.  6d.  lqr.  per  acre,  for  5a.  2r.  15rd.  of  land  ; 
what  was  the  entire  606t  \ 

22.  If  365^  days  make  a  year,  how  many  days,  hours,  etc., 
arc  there  in  .785  of  a  year? 

23.  What  is  the  cost  of  3  pieces,  of  cloth,  the  fir>t  containing 
id-,  at  $2.25  a  yard;  the  second,   12*5  yards,  at  S*>.50  a 

yard  ;  and  the  third,  8.8  yards,  at  $8.25  a  yard? 

24.  A  threi -sided  plat  of  ground  is  inclosed  by  a  railroad   on 
tide,  and  highways  on  the  other  two  sides ;  the  side  next  the 

railroad  is  4.1  rods  long,  and  the  other  two  sides  are  respectively 
•1  rods  and  .i)  of  a  rod  in  length  ;  what  is  thecostof  fencing  this 
pint,  the  fence  costing  S3.75  a  rod? 

It  a  boat  sails  8.7.">  miles  an  hour,  how  far  will  it  sail  in 
8. 1  1 1  < 

Sow   many  bins,  each  holding  37.5  bushels,  will  be  filled 
with  1687.5  boaheli  of  grain? 

27,    How  many  coats,  each  requiring  2.75  yards  of  cloth 
be  made  from  '.W>.7-'>  yards? 


1U 


UNITED    STATES   MONEV. 


28.  In  how  many  days  will  a  man   earn  $20.12."),  if  ho   earn 
$1.76  a  day? 

29.  How  many  square  feet  in  a  board  which  is  18.25  feet  long 
and  2.8  feet  pride! 

30.  Bought  a  load  of  straw  that  weighed  It.  2cwt.  3qr.  12£lb., 
at  $8  a  ton  ;  what  shall  I  pay  for  the  load  ? 

31.  Paid  $7,175  for  35  gall.  3qrt.  lpt.  of  vinegar;  what  was 
the  price  per  gallon  ? 

32.  If  a  pole  12.5  feet  long  casta  a  shadow  3.125  feet  at  12 
o'clock,  what  la  the  bight  of  a  steeple  that  casts  a  shadow  33.28125 

it  the  same  time? 

33.  What  n  the  cost  of  carpeting  a  room  that  is  16.5  feet  long, 
and  15  feel  widi •.  thr  carpet  costing,  $1.25  per  square  yard? 


UNITED    STATES    MONEY. 

177.    Tnited    States   Money,  sometimes  called  Federal 
Money )  is  the  currency  of  the  United  States. 


TABLE. 

10  Mills  (n 

i.)     make 

1  Cent, 

marked 

c. 

10  Cents 

u 

1  Dime, 

M 

d. 

10  Dimes 

u 

1  Dollar, 

CI 

$ 

10  Dollars 

H 

1  Eagle, 

Cents. 

II 

Mills. 

e. 

Dimes 

1 

— 

10 

Dollars. 

1 

=         10 

sa 

100 

,,dc. 

1      ~~~ 

10 

=       100 

— 

1000 

1     = 

:      10      = 

100 

sa     1000 

as 

lOOoo 

The  terms  eagle  and  dime  are  seldom  used  in  computation ; 
-  and  dollars  being  road  collectively  and  called  dollars,  and  dimes  ami 
cents  being  called  cents  ;  thus,  3  eagles  and  5  dollars  arc  called  $35,  and  4 
dimes  and  3  cents,  are  called  43  cent-. 


177.    What  is  United  States  Money?    Bepeat  the  Table.    Are  the  terms  eagle 
and  dime  much  used? 


I  ■!."> 

17**.    Tl      ■  n  -  ■       of  the  United  S  upon 

rules  for  i 

in   tbil  .I  al-o   manv   example*,   have   ah<  I 

:  but  th>-  importance  of  the  mbject  ju.-ti: 

ion  of  it. 

170.    A  ■■il'l.  silver,  or  other  metal,  stamped 

by  authority  of  tin-  Genera]  Government,  to  be  need  ai  money. 

1**0.    The  coins  aothorixed  by  our  Government,  and  stamped 

at  the  l'.  S.  Mint,  arc  the  following: 


Gold. 

Silver. 

Doulilc  Eagle, 

0,6ft 

Dollar, 

$1.00 

10*00 

Half  Dollar, 

Halt 

Quarter  Dollar, 

.25 

Quart' 

Dime, 

.10 

-Dollar  1' 

Half  Dime, 

.05 

One  Dollar, 

L00 

Three -Cent  Piece, 

.  of  Copper 

and  Nick<  I, 

ant, 

.01 

181.    Gold  and  aihrer,  for  <•< >ii.  hardened  by  b 

I  with  harder  and  cheaper  metals.     These  cheaper  m 

when  combined  with  the  gold  and  silver,  are  called  alloys. 

183*  Carat  is  a  term  used  in  indicating  the  purity  or  fine- 
ness of  gold.  If  a  piece  of  metal  is  pure  gold  it  is  said  to  be 
24  carats  fine  ;  if  r.  ]  of  it  are  gold,  and  the  remaining  ^  is  alloy, 

. 

1S3.     The  ttamdard   purity  of  gold  and  silver  coin  at  the 

DUN  metal  and  ^  alloy.      The  alloy  in   >il- 

iin  i>  pore  copper.     The   alloy  in  gold  coin  is  copper  and 

.    silver  not  l  the  oopp 

(a)  The  new  of  88  part.-  of  copper  for  12 

118.   0*  1  .iirnncy  of   the   U,  8.  band!     179.    Wliat  is  a  coin' 

180.    What  gold  coins  nre  authorized  by  our  Government '     What  rflr«f 

:   Mdl!     181.    What   is   aI!o\  ?     lor  wliat  used?     183.    For 
irat  used?    Pure  gold  is  how  many  carats  tine?     183.    Wlmt 
•ity  of  gold  and  silver  roin'     What  is  the  alloy  for  diver' 
What  for  gold?    What  part  of  the  new  cent  is  nickel? 

13 


40  IXITED    STATES   MON! 

m  1.  The  copper  cent  is  still  in  use,  but  is  no  longer  coined  at  the 
U.  S.  Mint. 

Note  2.     The  mill  is  not  coined. 

Note  3.  Other  pieces  of  money,  u  the  50-doOw  gold  piece,  the  half 
■ad  quarter  dollar  gold  pieces,  arc  in  u>e  to  KMM  extent,  but  arc  not  legal 
coin. 

pk  4.     The  greater  part  of  the  money  in  general  use,  OOMbtl  offtsafe 

Lilh,  which  are  niu-li  more  convenient  fat  most  purposes  than  gold  and 
silver. 

184.    The  weight  of  the  en  s  ltijik,  Troy.     The 

silver  dollar  weighs  412^  grains,  but  the  smaller  coins  are  not 
so  heavy  in  proportion  to  their  value;  thus,  the  lialt*  dollar 
weighs  only  192  grains;  the  quarter,  only  00  grains,  etc  The 
new  cent  Wtighfl  72  grains. 

Note.  These  standards  of  weight  and  purity  arc  regulated  by  Con- 
gross,  and*  may  l>c  changed  at  any  time. 

18t5.  In  this  currency,  the  dollar  \>  the  unit,  cents  and  mills 
being  deeimait  of  a  dollar;  thus,  $8.62  represent!  three  dollars 

and  BlXty-tWO  itfl  iour  dollars,  eight  cent-, 

and  live  mills,  etc. 

Note.  Figures  at  the  right  of  the  third  decimal  place,  npmtol  j«irtsof 
mills;  thus,  $5.3627  =  5  dollars,  36  cents,  i>  mills,  and  t7o"  of  a  mill. 

REDUCTION. 

186.  The  reduction  of  U.  S.  Currency  is  very  simple. 
Dollars  are  reduced  to  cents  by  annexing  two  ciphers  (Art.  62), 
and  to  mills  by  annexing  three  ciphers  ;  thus  $  I  =  400  cents  = 

Dollars  and  cents  are  reduced  to  cents  by  removing  the  deci- 
mal point;  thus,  $3.56  =  356  cents.     Dollars,  cents,  and  mills 

1^3.  Is  the  mill  coined?  What  of  other  pieces  of  money?  What  of  paper 
money?  184.  What  is  the  weight  of  the  eagle?  Of  the  silver  dollar?  Half 
dollar?  By  whom  is  the  standard  of  weight  and  purity  fixed?  185.  What  is 
the  unit  in  this  currency?  What  are  cents  and  mills?  What  are  figures  at  the 
right  of  the  third  decimal  place?  18G.  How  are  dollars  reduced  to  cents? 
Jlow  to  BriBsT  How  are  dollars  and  cents  reduced  to  cents?  How  dollars, 
cents,  and  mills  to  mill? ' 


117 


are  reduced  to  mitts  In  the  Bam  thus,  $5,468  =r 

mills. 

:.  Reduo  Aii  ;it<. 

•j.  Reduce  $8446  to  <•■  An  .  nte. 

.:.    i:    ;  :         .).45G  to  mills.  Acs.  8436  mills, 

l.  Reduce  $488  to  cents.    To  milN. 
...  ft  dncc  |6J  i  To  mitts. 

Reduce  $1.87G  to  mills. 

187.  Ctfnrt  are  reducrd  to  dollars  hy  pointing  off  two  d>  <{- 
mal  placet  (Art.  81).  Mills  are  reduced  to  dollars  by  pointing 
off  three  decimal  plans;  thus,  37G8  cents  =  $37.68  ;  37G8 
mills  =  So.768. 


7.  Reduce  5G4  cents  to  dollars. 

s.  Reduce  8692  mills  to  dollars. 

9.  Reduce  87G94  cents  to  dollars. 

1<>.  Reduce  76848  mills  to  dollars. 


Am.  $5.64 

>3.G92. 


I**.  Addition,  Subtraction,  Multiplication,  and  Division 
of  U,  S.  currency,  are  performed  precisely  as  the  correspond- 
ing operations  in  I><  dmaX  Fractions. 


ADDITION. 


$  8  7  6.5  4  2 
3  9  7. 1  8  8 
6  7  9.8  2  1 


3. 
6  4  8  7.3  3 
4  2  9  6.8  7 
•1  4.98 


Ex.  1. 
$  7  5.5  6  4 
2  4.8  7  6 
9  6. 1  45 

Sim,.  s85 

1.    Paid   $87.50  for  a  horse,   $145.25    for  a  pair  of   - 
?1  L25    for  a  wagon,  ami    $1~>.75  for  a  cart  ;  what  did  I  pay  for 
all?  An-.  $292.75. 

lit  a  hat  for  $  !.  .75,  a  vest  for  $5.25, 

ami  a  pair  of  boots  for  $."»  ;  what  did  I  pay  for  all  ? 


.  How  arc  cents  reduced  to  dollars.'  ll<>\v  mills  to  dollars?  188.  How 
are  Addition,  Subtraction,  Multiplication,  nnd  Division  of  V.  S.  Money  per- 
formed? 


148 


VNITi:i>    STATES    MONEY. 


SUBTRACTS  )\. 


From 
Take 

Kx.  1. 
$  4  8  7.9  6  1 
$2  6  8,7  8  8 

2. 

■;.8  7 

$47.43 

3. 

$8  6,4  8  5 
$  4  4.3  6  8 

Ans. 

$2  1  9.1  7  6 

4.  A  man  who  owed  SG99.GO,  paid  $1C4.60;  how  much  <1M 
he  -till  owe?  An 

5.  Bought  a  farm  for  $3G84.7"'.  and  Stock  and  tools  for  the 
farm  fur  $1  L7&25  :  how  much  more  did  I  pay  for  the  farm  than 
for  the  stock  and  tools  ? 

MULTIPLICATION. 

Ft    1  Ft    5 

Multiply     $3  4  8.7*65                             $3684.375 
By  2  5J  2^1     7 

13  950 80 

1743825  3. 

697530  $4386.942 

Ans.  S88586.310  369 

4.  If  12  gentlemen  have  $7497.84  apiece,  what  sum  have 
they  all  ?  Ans.  $89974.08. 

5.  If  45  persons  deposit  $346.25  each  in  a  savings  bank,  how 
many  dollars  arc  deposit 

DIVISION. 
K\.  l.  I;  $225  are  divided  equally  between  27  men,  what  sum 

will  each  receive  ? 

OPERATION. 

L>  7  )  $  2  2  5  (  $  8.3  3  $,  Ans.  Dividing  225  by  27,  g 

2  1  6  8  for  quotient  and  9  for  re- 

^  mainder.     Annexing  ciphers 

g  ••  and  continuing  the  division, 

as    in     Decimal     Fractions 

9  0  (Art.     172,    c),    we    obtain 

8J  $8.33£  for  the  share  of  each 

o  man. 


KONBY<  14'J 

2.   Divide  $69  I  18  men. 

Ans.  $3852.52. 
,40  into  21  equal  pari 

PRACTICAL  Examples. 

1*9.    To  find  the  cost  of  a  number  of  things  when 
price  of  one  thing  is  given. 

1.  If  applet  arc  worth  $2.50   per  barrel,  irbat  are  3  barrels 
worth  ? 

Three  barrels  are  worth  8  times  as  much  as  one  barrel,  .'.  3 
barrels  are  worth  S*-\50  X  3  =  §7.50,  An,      1 1  dice, 

BULK.     Multiply  the  price  of  one  by  the  number. 

2.  What  is  the  cost  of  9  barrels  of  flour,  at  $7.75  per  barrel? 

Ans.  $G9.75. 
Bought  25  sheep,  at  $&88  each  ;  what  was  the  cost  of  the 
flock? 

1.   Bought  18  yards  of  broadcloth,  at  $3,875  per  yard;  what 
WWi  the  cost  of  the  piece  ? 

5.  What  is  the  value  of  75  acres  of  land,  at  $37.50  per  acre? 

IOO.    To  find  the  price  of  an  article  when  the  cost 
of  a  given   number  of  articles  is  known. 

6.  When  eight  cords  of  wood  arc  worth  $44,  what  is  the  value 
of  1  cord  ? 

If  8  cord<   are  worth   $44,  one  cord  is  worth  £  of  $44;  and 
■.."»<».  Ana.      Hence, 

Rule.     Divide  the  cost  by  the  nu> 

7.  U  2  1  y.uds  of  broadcloth  cost  $93,  what  is  the  price   per 
yard  F  s7£. 

8.  Bought  37  pounds  of  butter  for  $8.51,  what  was  the  price  ? 

Ana,  23c 

189.    How  is  the  cost  of  a  number  of  things  found  when  the  price  of  one  ii 
kuowu?     190.    How  tli.-  price  of  oue  when  the  cost  of  a  number  is  known! 

13» 


150  i  XITED   STATES   MONEY. 

Note.  Price  is,  appropriately,  the  sum  asked  for  one  article;  thus,  when 
any  one  asks  a  flour  dealer  the  price  of  flour,  he  is  understood  to  ask  what 
lie  must  pay  for  a  single  barrel,  not  fifty  barrels,  nor  half  a  barrel,  nor  any 
quantity  except  one  barrel.  Hence  we  distinguish  between  price  and  cost,  or 
\>rice  and  value. 

'».   Bought  350  bbls.  of  flour  for  $302 G  ;  what  was  the  price  ? 

10,  Bought  a  farm  containing   1 2.3  acres  for  $6843.75;  what 
MM  the  price  per  acre? 

191.  To   fnul   the  quantity   when   the  cost  of  the 

quantity  ami   the   price  of  ono  arc  gi\ 

1  1.  At  $G  per  ton,  how  many  ton-  of  00*1  can  I  buy  for  $21  ? 
I  can  buy  as  many  tons  a9  $6  is  contained  timet  fin  $84  ■  illl(l 
:-  $G  =  4,  .-.  I  can  buy  4  Dons.     1 1  nee, 

r.     Divide  the  cost  by  the  price  of  one. 
12.   At   >:;  per  yard,  how  many  yards  of  cloth  can  be  bought 
for  $546?  An,  182. 

.50  per  acre,  how  many  acres  of  land  can  be  bought 
for  $  i 

11.  At   5G  cents  a  pound,  how  many  pounds  of  tea  may  be 
bought  for  $25.20? 

15.  A  drover  bought  oxen  at  $62.50  eaehj  how  many  oxen 
did  he  buy  for  $1562.50? 

192.  To  find  the  cost  of  articles  sold  by  the  100  or 
by  the  1000. 

l&  At  S1.50  per  100  feet,  what  will  842  feet  of  timber  cost? 

Had  the  price  been  $4.50  per 
foot,  the  cost  would  have  been  $4.50 
X  342  =  $1539;  but  unee  the  price 
U  $4.50  per  hundred  feet,  the  true 
multiplier  is  one  hundredth  part  of 
342,  viz.  3.42,  and  the  true  cost  is 
0  0,  Ans.  ,50  x  3.42  =  $15.39. 

•  90.  Meaning  of  price  T  Difference  between  price  and  cost,  or  price  and  value? 
191.  Rule  for  finding  the  number  of  things  when  the  cost  and  price  are  known* 
10*.    BxpbteEx.lt 


3.4  2 

1800 
1350 


151 

1   the  price  been  $4.5n  pet  thousand  feet,  the   true  multi- 
would  ha\  B48,  and  ti.  iM  have  Im.-h  $4.50 

X  .3 12  =  81.530.     Hence, 

Ki  :  the  quantity  to  hundreds  and  decimals 

of  a  hundred,  or  to  thousands  and  decimals  of  a  thousand,  as 
the  example  may  require  ;  tin  n  multiply  the  price  by  the  quantity, 
and  point  the  product  as  in  multiplication  of  decimal*  (Art. 
171). 

ml.     C  is  used  to  indicate  hundreds,  and  M  to  indicate  thousands. 

17.  What  cost  1200  feet  of  boards  at  $2.10  per  C? 

Ans.  $25.20. 

18.  What  cost  12514  feet  of  timber,  at  $13.50  per  M  p 

Aus.  $1G8.939. 
Note  2.    In  business  transactions  the  answer  to  Ex.  18  would  be  called 
8163.94.     In  the   remaining  examples  in   V.  S.  Money,  the  mills  in  the 
answers  will  he  omitted  if  less  than  5,  and  one  will  be  added  to  the  cents  if 
the  mills  are  5  or  more. 

19.  What  cost  20000  shaved  pine  shingles,  at  $6  per  M? 

20.  What  cost  13725  bricks,  at  $6.50  per  M  ? 

Ans.  $89.21. 

(a)  To  find  the  cost  of  articles  sold  by  the  ton. 

21.  What  cost  24401b.  of  hay,  at  $18.50  per  ton  ? 

OPERATION*. 

First  divide  by  2000  (i.  e.  point  off 
three  decimal  places  and  divide  by  2), 
to  reduce  the  WQlght  to  tons  and  deci- 
mals of  a  ton  ;  then  multiply  by  the 
price. 

In  multiplying,  the  50  cents  may  be 
used  decimally,  or  the  common  fraction, 
$2  2.5  7,  Ans.         \,  may  be  D8M,  U  in  the  operation. 

22.  What  cost  58481b.  of  coal,  at  SG.25  per  ton  ? 

Ans.  $18.28. 

193.    Rule  for  finding  the  eost  of  articles  sold  by  the  100  or  1000.     For  what 
U  C  used?    M  I    \\  i  timliug  the  coet  of  articles  sold  by 

the  tour 


2  )  2.4  4  0 

1.2  2 
18* 

61 

976 
122 

163  UXITl.b  i-Y. 

193.    To  find   the  cost  or  value  of  any  Dumber  of 

articles  when  the  price  is  an  aliquot  part  of  a  dollar. 

TABLE  OF  ALIQUOT  TARTS  OF  A  DOLLAR. 
50    cents  =  1  of  a  dollar,  20    cents  =  4,  of  a  dollar, 

te  =  I  of  a  dollar,  1 G  J  cents  =  |  of  a  dollar, 

25  'cents  =  \  of  a  dollar,  I2|  cents  =  \  of  a  dollar. 

23.  "What  cost  C  I  !'  cloth,  at  87)  cents  per  yard  ? 

OPERATION. 

$G_4  =  costof  Gh.l.  :.t   SI. 

3  2  =  cost  of  G4yd.  at        5  0  c,  or  h  of 
1C  =  cost  of  G4*vd.  at         2  5  c.  or 
_8  sb  cost  of  C4yd.  at        _l_2$c.,  or  £  of  25c. 

Ans.  $  5  G  =  cost  of  G4yd.  at        8  7  J  c. 

The  cost  at  $1  is  evidently  as  many  dollars  as  there  are  yards  ; 

the  cost  at  50c.  is  half  as  much  as  at  $1  ;  the  cost  at  25c,  half 

as  much  as  at  50c;  and  the  cost  at  12$c,  half  as  much  as  at 

Then  the  cost  at  50c,  at  25c,  and  at  12£c,  added,  gives 

the  cost  at  87  £c 

This  process  is  uuially  called  Practice,  for  which  we  have  the 
following 

Rule.     Take  such  aliquot  parts  (Art.  119,  Note)  of  the  num- 
ber of  articles  as  the  price  is  of  | 

2  1.   What  cost  48  barrels  of  apples,  at  $3,371  per  barrel? 

OPERATION. 

$4  8  =  cost  at  SI. 
_3 

8144  =  cost  at  $3. 

1  2  =rcot!  a;        .:'     c.  or  \  of 
G  =  cost  at       .12^  c,  or  £  of  25e. 

Ans.  $  1  G  2  =  cost  at  S3.3  7  }  c 

25.  What  cost  24  barrels  of  flour  at  $6.33$  per  barrel? 

Ans.  $152. 

193.    Rule  for  finding  the  cost  when  the  price  is  an  aliquot  part  of  a  dollar? 
What  is  this  process  called?    Name  the  most  convenient  aliquot  part*  of  a  dollar. 


153 


26.  What  co<t  is  11,.  of  raisins,  ti  *  pound? 

27.  wi  -i.  of  eali  r  yard? 

i  75  bosh,  of  apples,  at  :;:;>..•.  per  bushel? 

1  i<>  pair-  per  pair? 

ich? 

194.   To  find  the  cost  when  the  number  of  articles  is 

r Pepcid  by  a  compound  or  by  a  mixed  number. 
81.  What  oori  9a.  3r.  20nl.  of  land,  at  $40  per  acre  ? 

OPEKATI 

$4  0,  price  per  u« 
9 

$360  =  cost  of 

2  0  =  cost  of  2r.,  or  },:\. 

10  =  cost  of  lr.,  or  £  of  2r. 
5  =  cost  of  20rd.,  or  frr. 

$ 3  9  . '>  =  <ost  of  Da.  3r.  20rd.,  Ans. 

32.  What  corf  8>   shares  of    railroad  stock,  at  $108.50  per 
share  ? 

OPERATION. 

S  1  0  8.5  0,  price  per  share. 

8J 

ss  lis.nn  seosJ  of  8  shares, 

5  1.2  .">  =  cost  of  |  share, 

2  7.1  3  =  co-t  of  |  share, 

$  9  4  9.3  8   =  cost  of  8$  shares, 
This  process  is  also  called  Practice,  and  may  be  stated  thus: 

Multiply  the  price  by  the  entire  number  of  articles,  and  to  this 
ct  add  such  aliquot  purtt  of  the  price  as  the  fractional  part 
of  the  number  is  of  a  unit. 

33.  What  oust  3t.  16ewt.  Iqr.  201b.  of  hay,  at  $16  per  ton? 

1.16. 
84.  Whai  5c.  ft,  8ca.fi.  of  wood,  at  $6  per  cord? 

•  .  What  cost  24|  acrei  of  land,  at  $48.72  per  acre? 

«'JJ.  Mtag  the  cost  when  the  number  of  articles  ii  expressed  by 

uml  or  by  a  mix<.<l  ru 


154  UNITED   STATES   MONEY. 

195.    To  exchange  goods. 

86,  I  low  many  pounds  of  butter,  at  20c.  per  pound,  shall  be 
given  in  exchange  for  4  yards  of  cloth,  at  $2.87}  per  yard? 

Solution.     One  yard  o  .  1  yarda  cost  4  times 

$2.87(  =  $9.50.    Now  unce  tin-  price  of  the  butter,  20c,  is  £ 
of  a  dollar,  it  will  require  five  timet  Bfl  many  pouncKfl  of  \>. 
u  there  are  dollan  io  the  cost  of  the  cloth,  and  5  times  9.5  = 
47.5,  or  47.1,  number  of  pounds  of  batter  required,  I 

Dividing  $9.50  by  20c.  will  give  47.5,  or  47*,  the  same  result 
a>  before* 

Thii   exchanging  of  goods  is   usually  called  Barter.      The 
examples  are  solved  by  Analysis. 

.  How  many  pounds  of  sugt»i\  at    \'2\e.  per  pound,  may  be 
j-lit  for  3  bushels  of  corn,  at  87 f,c.  per  bushel  ?      Ans.  21. 
38.  How  many  cords  of   wood,  at  $5.50  per  cord,  shall  be 
given  in  exchange  for  a  barrel  of  flou**,  at  $7.50,  and  5  yai 
cloth,  at  $2.35  per  yard? 

BILLS. 

190.    A  Bill  of  Goods  is  a  written  ftatement  of  articles 
sold,  giving  the  price  of  each  article  and  the  cost  of  the  whole. 

Find  the  cost  of  the  several  articles,  and  the  amount  or  foot- 
ing of  each  of  the  following  bills. 

(1.)  Boston,  Jan.  1,  W 

Mr.  Abel  Snow, 

Bought  of  Jonx  Ada.m,«, 

2  5/5.  N.  O.  Sugar,  at'  9c. 

4  0lb.  M  iar,  18?e. 

6  lb.  Cheese,  «  12 

Sib.  But  «  2:; 

4  lb.  Raisins,  "  1  5 1 . 

2  lb.  Cream  Tartar,  «  4  5  c. 

$  1  3.8  4 


Received  Payment, 


John  Adams. 


195.    What  is  Barter?    How  are  examples  in  barter  solved?    196,    What  is 
a  Bill  of  goods? 


IW'ITLD    ST A. 

New  York,  Jan.  15,  1.^ 

mi. 

Bought  of  J  auks  Phil: 

•adcloth,          at  .">  0 
10                     Broadcloth,          u 

7  y ■>.  I          xrf,                    "  1 .2  5 

A  yd.  Black  Satin,                  "  4.5  0 


Received  Payment, 

.Jami> 

$  9  2.2  5 

i  Phillips, 

7fy  K.  Low. 

Philadelphia,  Mar.  1,  18G2. 
-  I  EWART, 

L861.                                    To  Holt,  Wilder  &  Co.,  Dr. 

June 
Oct. 

5. 

12. 

To 

u 
M 
M 

G  JFefater'*  Dictionaries,    at 
1  2  /)ay's  Algebras, 
8  6  ftftom                            " 

9  ^o//o  A'A/c*, 

SG.OO 
1.5  0 

2.5  0 

Received  Paymrut. 

$8  5.5  0 

S.  Dam 

For  Holt, 

Wilder  &  Co. 

(1.)  On  Mar.  1,  18G2. 

LP.   -ll   WETT, 

18»'.i.  Samuel  Palmbb,  Dr, 

Apr.    &    ZW6750J  4    $!2£0perM.    $209.38 

•■      175  M. 

May  IS      "  "'ink,       "       2  5.0  0^erJ/.  

$3  3  8.8  8 
-  'I. 
May    :».    By    3   T<m$  &%    erf  $15.5  0     $4  6.50 

5  0.0  0 
.  12.     «     4  Cbftb  irooiA«<         <■'•,,,, 

$  1  2  0.5  0 

Ild.mrr  fhu  &  P.  $218.38 

Received  Payi 

Sam i  i  i    1*  vi  ■ 


UNITED    STAT1 


Miscellaneous  Examples  in  U.  S.  Mom. 

1.  What  cost  3^  yards  of  ribbon,  at  56c.  per  yard? 

2.  What  cost  8  barrels  of  flour,  at  $7.62]  per  barrel? 

3.  If  4  cords  of  wood  c« I  t  $22.50,  what  i<  the  price  per  cord? 

4.  If   15  yards  of  sQk  OOSt   ^lG.-s?1,  what   is   the  pric 

yard  ? 

5.  If  a  Uterchml  depot  .50  in  a  bank  at  one  time,  and 
$487.75  at  another,  how  ranch  will  remain  after  he  has  with- 
drawn S17<;.:;7  and  S.J46.83? 

6.  A  merchant  bought  70  barrels  of  flour  for  $650  and  sold 
25  barrel  I  per  barrel,  and  the  remainder  at  $9.25  per 
barrel;  did  he  gain  or  lose?   How  much?     Ans.  Gained  $50. 

7.  What  nil  B7|  rodi  of  wall,  at  75c  per  rod  f 
Redoes  to  mills. 

'.).    Reduce  $6.18  to  mills. 

l<>.  Rednce  54598  cents  to  dollar-. 

11.  Reduce  47689  mills  to  dollar. 

12.  Bfy  farm  cost  $3725  and  my  house  cost  $1862.75  ;  how 
much  more  did  the  farm  OOSt  than  the  hoi, 

13.  A  gentleman  bequeathed  S750  to  each  of  his  3  sons,  and 
$500  to  each  of  his  4  daughters  ;  how  much  did  he  bequeath  to 
his  children  ? 

14.  Paid  $16.50  for  a  coat,  $4.25  for  a  vest,  $5.75  for  a  pair 
of  pants,  $3.50  for  a  hat,  $4.37^  for  a  pair  of  boots,  and  $12,621 
for  other  articles  ;  what  did  I  pay  for  all  ? 

15.  Divide  $113.7">  equally  between  7  men. 

16.  Paid  $68.75  for  flour,  at  $6.25  per  barrel;  how  many 
barrels  did  I  buy? 

17.  How  many  yards  of  lace,  at  62  £c.  per  yard,  may  be 
bought  for  $3.75  ? 

18.  What  cost  8725  feet  of  boards,  at  $12.50  per  M? 

19.  What  cost  8248  lb.  of  coal,  at  $6  per  ton  ? 

20.  What  cost  3a.  2r.  20rd.  of  land,  at  $48  per  acre  ? 

21.  How  many  pounds  of  sugar,  at  12£c  per  pound,  will  pay 
for  12  dozen  Qgp$  at  16^c.  per  dozen  ? 


l.~,7 

My  rea]  estate  ia  worth  i  and  my  persona)  i 

50  ;  trKat  am  I  worth  ? 
23.  an  1  carriage)  how  far  may  I 

2  1.  A  dnr?  r  bought  sheep  al  $8^7]  per  head  and  aold  them 
per  head,  and  gained  $87.50  by  the  transactions ;  how 

many  ibeep  did  he  buy  ? 

Bought  100  sheep  at  $& 875,  and  sold  them  again  at 
ha!  was  the  gain  per  head  and  total? 

Bought  20.5  tons  of  hay  at  $12£75  per  ton;  what  was 
tin-  coal  of  ilit*  whole? 

What  Si  the  value  of  67.75  acres  of  land  at  $02.50  per 
acre? 

28.  Paid   £1231.375   for  67.75  acres  of  land;  what  was  the 
price  per  i 

29.  Paid  $423 4.375  for  a  piece  of  land  at  $G2.50  per  acre  ; 
how  many  acres  were  bought? 

Bought  land  at  $62.50  pet  acre,  and  sold  it  again  at 
pes  acre,  thereby  making  $846,875;    how   many   acres  were 
lit? 
81.   Bonghl  87.75  acres  of  land  at  $62.50  per  acre,  and  sold 
the  lot  f<>r  $5081.25  ;  was  there  a  gain,  or  loss  ?  how  much  total 
and  per  acre  ? 

Bought  856.251b.  of  wool  at  37  lc,  which  was  manufactured 
into  cloth  at  an  expense  of  $62.50  ;  for  what  sum  must  it  be  Bold 
to  gun  $87.50? 

Bought  1  1.75yd.  of  sheeting  at  i4  cents  per  yd. ;  what  wai 
the  coat  of  the  piece  ? 

84    What  would   7$  bales  of  cotton   cost,  each   bale  weighing 

L75  per  cwt.? 
35.   What  cost  l-lyd.  2.jr.  3na.  of  cloth  at  $4.67  per  ell  French. 
the  ell  French  being  6qr.  ? 

86.  Bought  lbbl.  Hour  at  $12.50,  Sbush.  corn   at    8? 
11).  sugar  at  8Jc,  3gal.  molasses  at  37£c,   211».  tea  at  r.2.[c.,   Mb. 
at  lie,  t51ba>  rice  at    l]e<  and   41b.  butter   at   22c.;  what 
the  cost  of  the  whole?  Ans.  $21.76. 

What  cost  :;i    L5cwt  2.(r.  12.U1).  coal  at  $9.75  per  ton  ! 
14 


158  coMrorxD  numbers. 

38.  What  will  be  the  expense  of  papering  a  room  that  is  20 
feet  long,  15  feet  wide  and  8.5  feet  high,  a  roll  of  paper  being 
8  yards  in  length  and  £  of  a  yard  in  width,  and  costing  G2£c. 
per  roll  ? 

39.  Bought  133.5yd.  of  broadcloth  at  S3. 25,  and  sold  88yd. 
of  it  at  $3.88$,  50yd.  at  $3,875,  and  the  remainder  at  $3.60; 
Low  much  was  gained  by  the  transactions  ? 


COMPOUND  N  I   M1JERS. 
ADDITION. 

107.  A  Compound  Number  is  composed  of  two  or  more 
denominations  (Art.  8G)  which  do  not  usually  increase  decimally 
from  right  to  left;  consequently,  in  adding  the  different  denom- 
ination-, we  do  not  carry  one  for  ten,  but  for  the  number  it  takes 
of  the  particular  denomination  added,. to  make  a  unit  of  the  next 
higher  denomination  ;  thus  in  adding  Sterling  or  English  Money, 
oarry  1  for  4,  1  le  4qr.  make  Id.,  12d.  make 

nl  •Ji>s.  make  1 1". 

Ex.  1.  Add  6£  7s.  9d.  3qr.,  5£  12s,  lid.  2qr.,  27£  18s.  lOd. 
3qr.,  and  19£  14s.  8d.  lqr. 

operation.  Baring  arranged  the  numl 

■£        s.         d.    qr.  a>  in   tin4  margin,  the  amount  of 

6        7        9     3  the  right-hand  column  is  9qr.  = 

5     12      11     2  2d.    and    lijr.     Upon    the    lame 

2  7      18     10     3  principle  Bfl   in   addition  of  sim- 

19     14       8     1  pie  numbers,  the  lqr.  is  set  under 

c ,«,    rTa     T~a         a     7         the  column  of  farthings  and  the 
hum,  o  y     1  4        4     1  ..  .        ,       °        . 

2d.  are  added  to  the  pence  in  the 

mple,  making  40d.  =  3s.  and  4d.  Setting  the  4d.  under  the 
the  column  of  pence,  add  the  3s.  to  the  shillings  in  the  example, 
making  .Vis.  =  'J£  and  14s.,  and  so  proceed,  uutil  all  the  columns 
are  added. 

197.    Do  Compound  Numbers  increase  decimally?    Explain  Ex.  1. 


ADDITK 

19$.    The  j  of  procc:  ihc  samo 

as  in  addition  of  simple  numb 

To  odd  oompound  numbers,  » 

lit- 1 -i.      Write  the  numbers  so  that  each  denomination  shall 

orrupy  a  separate  c<dnm»,  the  lowest  denomination  at  the  right, 

Uft  in  the  order  of  their  values.     Add 

the   number*   in   the  lowest    denomination)  divide  the  amount    bij 

unher  it  takes  of  this  denomination  to  make  one  of  the 
higher,  set  the   remainder   Under  the   column,  and  carry  the  quo- 
I  •  the  next  column.     So  proceed  until  all  the  columns  arc 
added. 

199.    Proof.     The  same  as  in  Addition  of  Simple  Numbers 
(Art.  47). 


2. 

3. 

4. 

£ 

s.    d. 

qr. 

£             8. 

d. 

gal.     qt. 

]'t 

y  i 

4     7 

1 

3G     14 

9 

3     2 

1 

48 

9     0 

3 

18     12 

1  1 

1     1 

0 

10 

3     0 

1 

C4        8 

4 

1     3 

l 

36 

8  J 

3 

56     13 

C 

4     2 

1 

67 

4     8 

3 

42     12 

10 

2     0 

l 

i,    25  3 

9     9 

3 

219        2 

4 

13     2 

0 

,:;  158 

9     9 

3 

I  1.  In  writing  nml  adding  the  numbers  of  a  single  denomination, 
the  rules  of  simple  addition  must  he  ohserved ;  thus,  in  writing  the  pounds, 
in  Kx.  '2  m  units  under  units,  ami  tens  under  tens,  and  then,  having  added 
the  f;irthin:_rs,  peine,  and  shillings,  add  the  units  of  tht  pounda,  and  then  the 
tens,  as  in  addition  of  simple  numbers. 


5. 

6. 

7. 

lh.    oz.        dwt. 

IT- 

qt  pt.  gi. 

ft. 

r. 

rd. 

8        4     18 

22 

3 

3     1     3 

4 

1 

25 

6        4 

8 

4 

1     1     2 

6 

3 

1  6 

5     11      12 

18 

7 

2     0     3 

1 

2 

38 

6        8     11 

12 

•1 

8     1     0 

2 

0 

1  4 

2     10     16 

23 

9 

1     0     2 

6 

2 

24 

108.    I.                              1  of  0 

impoui 

id   Numbers?    P 

rinciplc 

1     1*9.    Proof! 

Numbers  of  a  tingle  denomination,  bow  written  and  added? 


160  COMPOUND   NUMBERS. 


8. 

9. 

10. 

lb.  oz.   dr.   sc. 

&r. 

bush 

.  pk.  qt. 

PI 

c.   c.  ft.  cu.  ft.        c.  in. 

8     4     G     2 

18 

G 

3     7 

1 

4     3     14     1 600 

G     9     2     1 

4 

8 

1     2 

o 

2    4       8       12  8 

2     10     2 

16 

9 

2     6 

1 

3     6     10        864 

8     8     3     2 

6 

4 

0     2 

1 

7     7        4       900 

11. 

12. 

1.  m.  wk.  (1. 

h. 

m. 

sec. 

t. 

cwt.     <|r.      ll>.         oz. 

2     3     4 

1  - 

40 

30 

6 

14     2     20        8 

3     3     6 

G 

20 

30 

4 

6     2     10        8 

5     1     2 

20 

30 

15 

;{ 

18     3     10     12 

8     3     0 

2 
13. 

28 

45 

4 

6     3     18        6 

14. 

circ.    s. 

o 

/ 

// 

yd.   qr.   na.  in. 

2      8 

20 

40 

50 

3     3     3     2 

1     4 

12 

18 

20 

8     2     3     1J 

G     6 

25 

50 

7 

6     3     10 

4     9 

29 

49 

59 

7     12     2 

1G. 

fur.      rd. 

vl. 

ft.      in 

i.     b.  c. 

yd.      ft.     in. 

1 

3 

2     10     1 

4       2        4 

2        4 

4 

2 

4     2 

3       1        7 

3        G 

5 

0 

6     2 

5       0        6 

1        3 

4 

2 
2 

7     0 
4    2 

rd.   4       2       7 

7     21 

3     U     1        0 

or  7     2  1 

2 

0     10     2 

or  3     1"     2        6 

Note  2.  A  fraction  occurring  in  the  amount  may  sometimes  be  reduced 
to  whole  numbers  of  other  denominations  ;  thus,  in  Ex.  15,  the  half  yard 
equals  1ft.  and  6in. ;  the  Gin.  ]>ut  with  the  4in.  make  lOin.  and  the  1ft.  put 
with  the  2ft.  make  3ft.  or  1yd.  Oft.,  and,  finally,  the  1yd.  pot  with  the  1yd. 
in  the  original  amount  gives  2yd.  The  answer,  when  reduced,  may  contain 
h  denomination  higher  or  lower  than  any  in  the  given  example  ;  higher,  as 
in  Ex.  16  ;  lower,  as  in  Ex.  17. 

199.  What  may  be  done  with  a  fraction  in  the  amount?  Explain  Ex.  15. 
Ex.  16.  Ex.  17.  May  the  answer  contain  a  higher  or  lower  denomination  than 
the  example?    How? 


ADDITI>  101 

17, 

a.     r.      r<l.       yd.        ft.  m.    fur.      nl.        ft. 

5  330     2  0       4  4     7     3  9     1  G 

6  2     1  I     1 7       7  3     G        8     12 


12     2 

3     1 

73     2 

in. 

or  IS     2 

3     17       8 

108 

19. 

20. 

t.     cwt 

qr.      lb. 

oz. 

dr. 

lm-li 

pk.  Mt. 

pi 

4        6 

a   so 

8 

i  a 

1 

8    7 

1 

2     1  1 

3        5 

7 

4 

1 

8     I 

0 

3        8 

1     1  G 

1  2 

8 

5 

0     G 

1 

1         7 

0     24 

4 

4 

3 

3     3 

1 

9     19 

8        l 

l  D 

5 

G 

1     0 

0 

4        G 

0        0 

4 

15 

5 

2     5 

1 

21.  Bought   4  pieces  of  cloth,  measuring  Gyd.  3qr.  lna.  2in., 
k\.  lin.,  25jd  Iqr.  2na.  2in.,  and  14yd.  3qr.  2na.  lin.; 
Miuch  cloth  did  I  buy  ? 

A  farmer  raised  in  one  field  21bush.  3pk.  7qt.  lpt.  of 
tl  ;  in  another,  48bu>h.  2pk.  lpt.;  in  another,  28bush  Gqt. ; 
and  in  another,  75bosh.  lpk.  oqt.  lpt. ;  how  much  wheat  did  he 
in  Ihe  l  BeMe  ? 

A    planter  sold  cotton   at    rarioOS   tim<'<,   as  follows:  2t. 

:.  2qr.  l-' Ul>..  6t  lewt  Iqr.  Gjlb.,  8t   I9ewt  8qr.  18|lb., 

16t.6ewt.8qr.  12111..,  and  16t8qr.  181b. ;  what  did  he  sell  In  all? 

2  1.  What  h  the  nun  of  l  la.  Jr.  80rd  85yd.  8ft.  72in.,  87a. 

tad.  80yd.  6ft.  86ul,  50a.  lr.  I8rd.  25yd.  2ft.  108in.,  and 

.  85id  25yd.  8ft.  72m.? 

25.   A- id   8ci  .  2circ  Us.  2.".°  20'  SO",  5circ. 

4«.  8  .  and  6eirc  10s.  1"°  lo'  io"  together. 

traveled  85m.  6fbr.  18rd,  5yd.  in  one  ihy,  42m. 
!.  the  next  day,  87m.  5for,  82rd.  4yd.  the  next, 

and  15m.  7fur.  2  Ird.  8yd.  the  next  ;  how  far  did  he  travel  in  the 
4  da; 

27.  A  blacksmith  bought  It.  18cwt  8qr,  201b.  of  iron  at  one 
time,  St  15ewt  8qr.  121b.  at  another  time,  St.  6cwt  Itjr.  181b. 
at  another,  sad  8t  ScwL  2qr.  101b,  at   another;  how  much  did 

iv  in  all  } 

11- 


L62  COMPOUND    NUMBERS. 


SUBTRACTION. 


200.  The  principle  is  like  that  of  subtraction  of 
simple  numbers.     Hence, 

To  subtract  compound   numb* 

Rule.  1.  Write  the  less  quantity  under  the  greater,  arranging 
the  denominations  as  in  addition, 

2.  Beginning  at  the    right,  take    each  denomination    of  the 
subtrahend  from  the  number  above  it,   and  set   the   r 
beneath. 

8.  If  any  number  of  the  subtrahend  is  greater  than  the  num- 
ber above  it,  add  to  the  upper  number  as  many  as  it  takes  of  that 
denomination  to  make  one  of  the  next  higher,  and  take  the  subtra- 
hend from  the  sum  ;  set  down  the  remainder,  and,  considering 
the  number  in  the  next  denomination  in  tlie  minuend  one  less, 
or  that  in  the  subtrahend  one  greater,  proceed  as  before. 

201.  Proof.    As  in  subtraction  of  simple  numbers  (Art.  53 ). 
Ex.  1.  From  8£  6s.  9d.  3qr.  take  2£  1  .  5d  lqr. 

OPERATION. 

£      8.    d.     qr. 

Miii.,    8     6    9     3  Only  the  Island  2d  sections  of 

.!>.,     2     4     5     1         Ihfl  rule  apply  to  this  example. 

Km.,  6     2     4     2 

Proof,  8     6    9    3 

2.  From  9£  fe.  LOd.  lqr.  take  t£  17a.  2d  8qr. 

operation.  As  8qr«  cannot  he  taken  from* 

£       s.     d.    qr.  lqr.,  borrow  OOQ  of  tlie  1<>«1..  r«- 

Min.,    9       6    10     1  it  to  farthings  and  add  it  to 

Sub.,    2    17      2    3  the  lqr.,  giving  5qr.j  then  aaj) 

Rem#    (j      9      7     2  tlu!n  ""lr"  I,':iu'  -'lr'     ^'ou'> 

*  as  one  of  the  lOd.  has  been  em- 
Proof,  9       6    10     1         ployed,  say  2d.  from  9d.,  or,  «  hat 
is  practically  the  same,  3d.  from 
lOd.  leave  7d..  and  so  proceed  through  the  example. 

200.    Rule  for  subtracting  compound  numbers?     Principle?     201.     Proof  f 
Explain  Ex    1.     Ex.  2. 


IN 

The  form  of  the  minuend   ni:iv  1  1  and  the  work  per- 

formed U  follow.   (All.  D 

SECov 

£         8.         d.     qr.  £       8.       d.     qr. 

Min.,   9       G10     1)(8     2G9     5 


«- 


Sub.,    2    17  2     3j  — (2     17     2     3 

K.  -in.,  G9  7     2     =     G        972 

3.  1. 

t.          cwt.  qr.      11).  oz.      dr.                          lb.   oz.   dr.    sc.      gr. 

From    12        8322  G     1  5                   6     4     3     118 

Take        3     19     2     18  8     12                  2     3     C     2     1  2 

Bern.,       8        9     1        3  14        3                  4     0     4     2        6 

f,   12        8     3     22  G     lo 

5.  G. 

yd.    qr.    na.    in.  1.    m.   fur.    rd.    yd.     ft.    in. 

From    16     12     1  62427*518 

6     3     12  222     35     225 


8. 


a.      r.      rd.  yd.      ft.        in.                                  pal.     qt.    pt    -i 

From     6     I     ■_'  5  3  0     4     134                         14     2     0     3 

Take      13     39  5     8     140                           5312 

'.'.  10. 

lb.        oz.    dwt.  gr<                                          c.     eft.   cu.ft.     cu.in. 

Min.,  G   5  15  25  4  15  1727 

S.il..,  3  10  1  2  y  2  3  4  7   5    169 

2   7    2  23  

6   5  15  22 

11.  12. 

bmh.     pk.  <]t.  pt.                wk.  d.   li.   m.       sec 

L  6  1              3  4  23  45 

2  4  8  7  1              1  G  1 G  3  0  4  5 


201.     Kxplain  2d  operation  in  Ex.  2. 


1G4  COMPOUND   XiMl  : 

202.     Sometime*,  as  in  the  following  exai  is  neces- 

sary to  borrow  two  of  the  higher  denomination  of  the  minuend 
in-trad  of  one;  but  in  all  such  cases  we  must  carry  too  to  the 

in-xt   term  of  the  subtrahend  ;  i.  e.  ice  must  PAT  at  much  as  tee 
no  into  w. 

13. 

rd.     yd.    ft.    in.  b.c.  rd.       yd. 

i    i  a  o   a   8 

Take        3    5     2    8     2 


1  )       (  l  0     l  0 
2|  —  \ 


ft. 

in.     b.  c. 

4 

17     4 

a 

8     2 

Rem.,       7     5     2     9     2    =        7        5     2        9     2 
Proof,    120261     =     10104174 


From 

Take 

a. 
7 
1 

r.       rd. 

2  0      0 

3  39    30 

ft. 
5 
8 

14. 

in. 
124 
143 

Rem., 

5 

1    39    29^ 

5 

1  25 

Proof, 

Min., 

Sub., 

7 
m. 

a 

2  0       0 

;o. 

fur.   rd.   vd.   ft 

3  7     0     1 

b  b  a 

0 

124 

12  4)    (6  1  7 
1  1  8  )  ~  X  1  3  3 


L  It   in. 

8  60  9  196 

9  3  0  8  148 


=  5  1  39  30  1   53 
=  6478G0  9  19G 

1G. 

circ.  deg.      m.     fur.      rd.   yd.   ft. 

7        0        0     1        0     0     1 

2276933951 


Krim     3     G     0     5     2  

Proof,    G     3     7     0     1 

30ft.    To  find  the  time  between  two  dates. 

17.  What   i-  the  difference  of  time  between  July  lo,  1857, 
and  Apr.  25,  18  Am.  lyr.Dm.  10d. 

FIRST    OPERATION*.  SECOND   OPERATION'. 

vr.         in.      (1.  vr.         m.      d. 

Min.,      18  62     4     2  5)        (186  1     3     24 
Sub.,      1857     7     15 }  or  X 1856     6     14 

Rem.,  4910=  4910 

202.    What  id  said  of  borrowing  tico?     Explain  Ex.  13.     203.    How  many 
modes  of  finding  the  time  betweeu  two  dates?    What  are  they? 


<\irlirr  fn> 
mth.     In  tin  .:ioii,  the  number  of  the 

:  the  month,  is  used  ;  in  the  iscood,  the  Dumber  of 
i  dayi  net  Aow  dapmi  lines  the  commencement  of  the  cinis- 

ra,  is  used.  cm  give  the  same  rc.-ult,  l.ut  As  yirs/  is 

1  I.   How  long  from  the  battle  of  Waterloo,  June  18,  181 
the  death  of  Napoleon,  May  .">,  1mm  ?      An-.  5yr,  lOitt.  I7d. 
19.  How  long  from  tin.'  battle  of  Lexington,  Apr.  19,  177.'». 

to  tin-  surrender  of  Cornwallis,  Oct  19,  1781  ? 

I  low  long  from  tin*  inauguration  of  Washington,  Apr.  30, 
tie  of  New  Orleans,  Jan.  8,  1815  ? 
•J  1.   How  long  from  tin-   1  >< claration  of  Independence,  July  4, 
I77t>,  to  the  present  time? 

Daniel   Webster  was  horn  Jan.   18,  1782,  and  died  Oct. 
ge  did  he  die? 
ote  given  July  6,  U  |  aid  Sept  9,  1861  ;  hovr 

vraa  it  on  interest? 
J  1.   Find  tin-  time  from  Apr.  4,  1857,  to  Dec.  12,  18< 

Find  tin-  time  from  Dec.  16,  1839,  to  Mar.  26,  1848. 
26.  Find  the  time  from  Nov.  13,  1816,  to  May  12,  1841. 
•J  7.   Find  the  time  from  June  21,  1842,  to  Feb.  20,  1860. 

BZAXPLBfl  EH   Ai»i»itk»n   and  Subtraction. 
I.  A  fanner  raited  I50tmah.  8pk,  4qt  of  oats.     Having  sold 

BObosh.    -jik.  and    BSed    27bosh.    lpk.    lijt.,   how   many    ha>    lnj 

remaining?  Ana.  7Sbosh. 

Baring  a  journey  of  127m.  4fur.  lOrd.  to  perform  in  3 
I  travel  48m.  2fbr,  6rd  the  nml  day,  and  54m.  4rd  the 

1  ;   how  tar  hum  I  travel  on  the  third  d 

I   have  one  piece  of  land  containing    17a.  8r.  25rd.  and 

another  contain.  '•.  l.">rd.  ;   how  much   land  shall    I    have 

•  Uing  87a. 
4.  From  the  mm  of  8bnsh.  8pk.  2qLlpt.and  lObnsh.  2pk. 

7<jt.  lpt.,  take  the   difference   between  5  ll>u>h.   1  j>k.  8qt   lj't.  and 

49boah.  Spk.  2qt  l;  .  Idbnah.  Iqt 

203.  Wliich  i>  preftnblet     How  many  days  rrc  considered  a  mouth! 


166  COMPOUND   NUMBERS. 

5.  From  the  sum  of  5rd.  lyd.  2ft.  4in.  lb.c.  and  4r<l.  2yd.  lft. 
!>in.  2h.<\,  take  the  difference  between  lOrd.  5yd.  2ft.  Tin.  2b.c. 
and  lid  lyd.  1ft.  Gin.  An?,  lb.c 

G.  From  a  piece  of  silk  mot  Wiring  10  yd.  lqr.  3na.  2in.,  there 
were  cut  3  dresses,  the  first  measuring  15yd.  8qr.  lna.  lin.,  the 
second  14yd.  3qr.  3na.  lin.,  and  the  third  14yd.  2qr.  3na.  2in. ; 
what  remnant  remained? 

7.  B  sold  an  ox  which  weighed  16cwt  lqr.  lolb.,  and  2  cows 
thai  weighed  6cwt  lqr.  10  lb.  arid  5cwt  3qr.  201b.;  also  2  swine, 
that,  weighed  4e\vt.  3qr.  181b.  and  3cwt.  3qr.  21  lb.      How  mucli 

more  beef  than  j>ork  did  be  tell? 

8.  If  from  2  casks  of  wine,  containing  63gal.  3qt.  Ipt.  ogi.  and 
56gal.  2qt.  2gi.,  there  be  taken  75gal.  2qt.  lpi.  ogi.,  how  many 
gallons,  quarts,  etc.,  will  remain? 

'.'.  From  a  mass  of  silver  weighing  471b.  8oz.  16dwt.  22gr., 
a  >ihersmith  made  48  spoons  weighing  71b.  8dwt.  M .nr.  and  a 
Cake-basket  weighing  31b.  6o*.  Bdwt  l"»Lrr. ;  how  much  silver 
remained  in  the  mass? 

MULTIPLICATION. 

*<20  I.  In  the  multiplication  of  both  simple  and  compound 
numbers,  the  multiplier  is  always  and  necessarily  a  simple 
abstract  Dumber;  for,  to  attempt  to  multiply  by  a  concrete  num- 
ber, e.  g.  4  miles  times  10,  is,  in  the  highest  degree,  absurd, 
though  it  is  perfectly  proper  to  >ay  10  times  4  miles.  The  pro- 
duct is  of  the  same  kind  as  the  multiplicand;  for  repeating  a 
number  does  not  change  its  nature, 

£03.  The  principle  is  the  same  as  in  multiplication 
of  simple  numbers.     Hence, 

To  multiply  a  compound  by  a  simple  number, 

Rule.  Multiply  the  lowest  denomination  in  the  multiplicand, 
divide  the  product  by  the  number  it  takes  of  that  denomination 

204.    "What  is  the  multiplier  in  all  cases?    What  the  product?    205.    Rule? 
Troof  ?    Explain  Ex.  L 


MlLTIl'LICATI  1<M 

ike  one  of  the   next  higher,  set   down    tki  mnaimln-,  odd 
tlie  quotient  to    the  product    of  the    next    doiomination,  and   so 
cd. 

Ex.  1. 

£     s.    d.    qr.  First,  7  times  3qr.  =  21  qr.  = 

Multiply    4    6    8    3        Sdandlqr.  j  write  the  lqr.  nn- 

7        &ta  the  farthings,  and  then  Bay  7 

In*  30    7     1     1        timesSd  =.^l.,an<l  r.l.  added 

give  61d.  =  os.  and  Id  , 

Note.     Multiplication  and  division  prove  e*ch  other.     It  is  profitable  to 
teach  reverse  operations  simultaneously. 


rd.  yd. 
Multiplicand,              5     3 
Multiplier,         fur 

Product,              1     4     5 

2. 

ft.    in. 
1      4 

1   10 

b.c, 

1 
8 

2 

gal  qt 

6     2 

47     0 

3. 
.  pi 

1 

0 

pi- 

7 

1 

4. 
lb.  oz.     dwt. 
Multiplicand,  4     6        8 
Multiplier, 

8T. 

20 

4 

5. 
lb.     oz.     dr. 
2      10      6 

se. 
2 

gr. 

15 

6 

Product,        18     1     15 

8 

6. 
t.     cwt.    qr.      lb.        oz. 
3     15     2     24     15 

dr. 
8 
8 

< 

ire. 
o 

7. 

vd.    qr.  na. 
6     2     3 

in. 
2 
9 

8. 
wk.    d.     h.      m.      sec 
1     2     4     45     59 
3 

9. 
s.       °         ' 
8     20     30 

ff 

25 
10 

10. 
gal.  qt    pt.        gi. 

8    3     1        2 
12 

11. 

bush.  pk.   qt. 

8     3     7 

Pt. 

1 

1  1 

J  68  OOMTOUXD    N 

12.  13. 

c      eft.  cu.fi.      rn.in.  a.     r.       nl.       nl.     ft.        in. 

12    7     15     1725  7    3    39     30    8     I 

4  4 


11.  The  J,yd.  in  the  pr<»- 

a.       r.       nl.       y.i.       ft.  (\\w\  equals  4j(t,  i.e. 

8       2       24258  4ft.  72in.:  the  4ft.  pot 

1  2  with  the  6ft.  make  loft., 

43     3     18  7^      6      in  or  lyd.  lft.;  and.  tinat- 

or,  4  3    3     18  8       1  lT*    putting    the    lyd. 

with  the  7yd.  . 
and  the  whole  product,  .-.,  is  43a.  3r.  18rd.  8yd.  lit.  72in.,  -\ 

15.  16. 

m.  fur.  rd.  vd.   ft.    in.   b.c.  *.     r.      rd. 
2     3     3    *4     1     6     1  7     2     2  0     I 
7  9 

17.  Bought  5  loads  of  wood,  each  BMMUiing  lc.  5c.  ft.  8cu.fL, 
at  S6  per  cord;  what  was  the  quantity  bought  and  the  00 

.\h..le?  Ans.  8c.  3c.  ft.  8cu.ft.;  %l 

18.  If  I  .-hip  sail  2°  3"  t  day,  how  far  will 
8  daj 

Multiply  8m.  Cfur.  12rd.  3yd.  2ft.  Gin.  lb.c,  by  6. 
20.  If  a  man  travel  25m.  6fur.  25rd.  per  day,  how  far  will  he 

1  in  9  da; 
31.  If  the  crop  of  hay  on  1  acre  is  2t.  15cwt.  2qr.  12^  lb., 
what  will  bo  the  crop  on  10  acres  ? 

What  cot  7  \  tldfl  of  cloth,  at  lo>.  Gd.  3qr.  per  yard  ? 

23.  How  much  wine  in  o  casks  containing  28gal.  3qt.  lpt.  2gL 
each? 

24.  Multiply  9m.  7fur.  8ch.  Bid  15 H  Sin.  by  8. 
Multiply  Scire  5a,  85*  K  2.V  by  9. 

£06.     To  multiply  by  a  composite  number  : 

BULB.     Multiply  by  the  factors  of  die  multiplier  (see  Art.  6l). 

205.    Explain  Ex-  14.     206.    Rule  when  the  multiplier  ia  composite? 


169 

26.  Multiply  111..  *oz.  lGdwt.  20gr.  b\ 

11).  oz.    dwt.     gr. 

Multiplicand,                             4  8     16    20 

l~t  Factor  of  Multiplier,  8 

Partial  Product.                            37      10      14      i~6 
2d  Factor  of  Multiplier,        9 

net.  34  1        0     12        0 

»7.  Multiply  7£  6s.  8d  2qr-  by  54. 
Multiply  8bash.  8pk.  6qt  lpt  by  81. 

29.  Multiply  Gib  4£  75  29  6gr.  by  49. 

207.    To  multiply  when  the  multiplier  is  large  and 
not  composite. 

30.  Multiply  8i  4cwt.  2qr.  61b.  8oz.  4<lr.  by  23. 

~T    OPERATION. 

t.       Art   qr.       lb.       oz.       dr. 

3        4     2        6        8        4  Multiplicand. 


II     11     3     2  0        9     1  2  =  7  times  multiplicand. 
3 

G7     15     3     11     13        4  =  21  times  multiplicand. 
6        9     0     13        0        8=     2  times  multiplicand. 

74        4     3     24     13     12  =  23  times  multiplicand,    Ans. 

First  multiply  by  21,  i.  e.  by  7,  and  that  product  by  3  ;  then 
add  twice  the  multiplicand,  and  thus  multiply  by  23. 

SECOND    OPERATION. 

t.   cwt.  qr.       lb.       oz.       dr. 
3     4     2        6        8        4  Multiplicand. 
6 


1  '.»     7     1     14        1        8  =  6  times  multiplicand. 
4 

7  7     9     2        G        6        0  =  24  times  multiplicand. 
3     4     2        6        8        4=     1  time  multiplicand. 

74     4     3     24     13     12  =  23  times  multiplicand,  Ans. 

II.  re  we  multiply  by  24,  i.  e.  by  6  and  4;  then  subtract  the 
multiplicand. 

15 


170  coMpi.u  m»  en 

The  foregoing  plan  may  be  indefinitely  modified ;  hence  thia 
general  direction : 

Multiply  by  two  or  more  numbers  whose  product  is  nearly  the 
multiplier,  and  add  to,  or  subtract  from,  the  product  such  numbers 
as  the  case  may  requ 

31.  Multiply  15yd.  2qr.  lna.  by  17. 
:l\  .Multiply  27gal.  lqt.  lpt.  2gi.  by  43. 
33.  What  is  the  cost  of  753  acres  of  land,  at  4£  10s.  8d.  2qr. 
per  acre  ? 

OPERATION. 

£         8.         d.       qr. 
4     10        8 

10 


4  5        7        1        0  =  cost  of  10a. 
10 


45  3     10     10        0  =  cost  of  100a. 

7 


817  4     15     10        0  =  cost  of  700a. 
2  2  6     15        5        0  =  cost  of  50a. 
1  8     1  I        1        2  =  cost  of      3a. 


8  415        3        4        2  =  cost  of  753a.,  Ans. 

Multiply  by  100,  i.  e.  by  10  and  10;  then  multiply  the  cost 
of  100  acres  by  7,  the  cost  of  10  acres  by  5,  and  the  cost  of  1 
acre  by  8,  which  will  give  the  cost  of  700,  50,  and  3  acres, 
erally  ;  finally,  add  the  cost  of  700, 50,  and  3  acres  together,  and 
thus  find  the  cost  of  753  acres,  the  answer. 

34.  If  1  acre  of  land  yield  54  bush.  3pk.  6qt.  lpt.  of  corn, 
what  will  0  18  acres  yield  ? 

35.  If  a  man  travel  33m.  6fur.  35rd.  5yd.  2ft.  llin.  each  day, 
how  far  will  he  travel  in  313  days? 

36.  If  a  ship  sail  2°  40'  30"  each  day,  how  far  will  she  sail  in 
127  days? 

37.  How  much  wine  in  157  casks  if  each  cask  contains  53gal. 
Sqi,  lpt.  2gi.? 

«07.    Mode  of  multiplying  when  the  multiplier  is  large  and  not  composite! 


KULXXPU  IT  I 

208.    To   find   the   difference  of  the  time  of  day  in 
two  :  •   the  Btme  absolute  moment  of  time,  when 

the  longitude  of  each  place  is  known. 

i  appears  to  go  from  east  to  west  round  the  partly 

:.  100),  in  24  hours,  it  appears  to  go  ^  of  3G0°,  viz. 

\")°  in   1  hour,  and,  consequently,  1°  in  ^  of  1  hour,  viz.  4 

minutes  and  1'  of  di-tanee  in  ^  of  4  minutes,  viz.  4  second?. 

the  following 

TABLE   OF  LONGITUDE  AND   TIME. 


360° 

of  longitude 

=   24  hours,  or  1  day  of  time, 

15° 

of  longitude 

=      1  hour  of  time, 

1° 

of  longitude 

=      4  minutes  of  time. 

1' 

of  longitude 

=     4  seconds  of  time, 

1" 

of  longitude 

=  j^of  a  second  of  time. 

// 

77        2       48 
4 


38.  When  it  is  12  o'clock,  noon,  at  Washington,  what  time  is 
it  at  London,  Washington  being  77°  2'  4#"  west  of  London  ? 

operation.  Since  l"  of  longitude  makes 

a  difference  of  ^y  of  a  second 

of  time,  48"  of  longitude  give 

y>0a  =  3£sec.  of  time,  and  for 

5h.     8m.  1  l£sec.  Ans.         a  like   reason   8!  of  longitude 

give  8sec.  of  time,  which  added 
to  the  3&sec  previously  obtained,  give  lljsec,  and,  final  h . 
of  longitude  give  4  times  77  =  308m.  =  oh.  8m.  of  time; 
.'.  the  difference  in  time  between  London  and  Washington  is  5b. 
8m.  lllsec.,  and  as  London  is  farther  east  than  Washington,  the 
hour  of  the  day  is  later  in  London  than  in  Washington,  i.  e.  it  is 
8m.  ll£sec  past  5  o'clock  in  the  afternoon  at  London  when  it 
is  noon  at  Washington.     Hence, 

RULE.     Multiply   the   difference   of   longitude,   expressed  in 

degrees,  minutes,  and  seconds,  by  4,  and  the  product  trill  be  (he 

nee  in  t>  ued  in  tninutes,  seconds,  and  SOths  of 

208.    Ilow  far  does  tlM  MB  appear  to  move  in  one  hour'   Wliich  way'    Give 
the  table  of  longitude  and  time.    Rule  for  finding  difference  in  time  of  two  placet 
<•  Iragttn4i  <>t  m 


172  COMPOUND   NUMBERS. 

ik.  1.  The  place  most  easterly,  has  its  hour  of  the  day,  at  a  piven 
moment,  latest ;  i.  e.  the  day  bcgUM  first,  noon  comes  first,  and  the  daj 
closes  first  at  the  place  most  easterly. 

39.  The  longitude  of  Boston  ifl   71°  4'  9"  west,  and  that  of 
liin-ton   is  77°  2' 48"  west ;  what  is  the  difference  in  the 

time  of  the  two  places,  and  what  time  is  it  in  Washington,  at  3 
o'clock,  P.  M.,  in  BosttO  ? 

subtraction,  the  difference  of  longitude  is  found  to  be  5° 
58'  3'.> ',  .-.  (he  ffiAueuee  in  time  is  2."'»m.  54gsec,  and  at  3  in 
Boston  it  is  36m.  and  5$sec.  pmst  1  in  Wellington,  Ans. 

40.  The  longitude  of  Paris  is  2°  20'  15"  east,  and  that  of 
New  York,  74°  0'  from  Greenwich ;  what  is  the  differ- 
ence in  time  in  the  two  places?        A  •.  Ans. 

Since  Paris  is  in  east  longitude,  and  Hew  York  in  west,  theii 
difference  in  longitude  is  found  by  adding  2°  2(/  15"  to  74°  (K  3". 

41.  What   is    the   difference   in    time   between    Philadelphia, 

igitude,  and  Chicago,  87°  35'  west  longitude? 

42.  What  is  the  di:"  in  time  between  New  Orleans, 
90°  7  t  longitude? 

43.  What  is  the  difference  in  time  for  90°  in  longitude  ? 

DIVISION. 

209.  Here,  as  in  the  three  preceding  sections,  the 
principle  is  the  same  as  in  the  corresponding  operation 
in  simple  numbers.     Hence, 

To  divide  a  compound  by  a  simple  number, 

Rule.     Divide  the  highest  denomination  of  the  dividend,  and 

set  down  the  :  if  there  is  a  remainder,  reduce  it  to  the 

lower  denomination  ;  to  the  restdt  add  the  given  quantity  of 

that  denomination,  and  divide  as  before,  setting  down  the  quotient 

and  reducing  the  remainder,  etc. 

808.  Which  has  the  hour  of  the  day  latest,  the  most  easterly  or  most  west- 
erly place  I  ■  difference  in  longitude  found  when  one  place  is  in  east 
and  the  other  in  west  longitude?  209.  Rule  for  dividing  a  Compound  by  a 
Simple  Number?    Principle? 


DH  it; 

.  l.  Divide  W£  7a,  1,1.  lqr.  !• 

OPKi:  un»\. 
X        s.     (1.    qr.  at  of 

7)30    7     1     1  nderof2i 

"~7    g    g    «    a  reduced  to  shillings  and  added 

17-.,  which,  divided 
by  7,  give  u  qnotient  of  6s. 
30    7     l     l.  Proof.        and  a  remainder  oi 

2.  Divide  lfur.  9rd.  2yd.  Oft.  Oin.  lh.c.  by  5. 

Aiw.  9rcL  iv.i.  2&  Gin.  2b.c. 

3.  Divide  20gal.  2qt.  Opt.  2gL  by  7. 

2gal.  3qt-  lpt-  2gu 
l.   Divide  181b.  loz.  15dwt  8gr.  I 
5.  Divide  171b.  5J  13  1&  1< 
G.  Divide  80t  ocwt.  3qr.  24B>.  12oz.  by  8. 

7.  Divide  (  'yd.  2qr.  8na.  by  9. 

8.  Divide  8wk.  Gd.  1  lh.  17m.  57sec  by  3. 
:•.   Dfvid  4'  10"  by  10. 

10.  Divide  I07gal  Iqtby  12. 

11.  Divide  Wbosh.  Bpk.  2qt.  lpt.  by  11. 

12.  Divide  51c.  7r.it.  L5ca.ft.  171Gcu.in.  by  4. 

13.  Divide  16a.  lyd.  1ft.  7<>in.  by  2. 

1  1.  Divide  87t  L2cwt  8qr.  51b.  10oz.  4dr.  by  9. 
15.   Divide  71a.  8r.  1  trd.  8yd.  lit.  72in.  1.;. 
L&  If  9  Bilvez  Bpoons  weigh   Lib.  l<»z.  17dwt  12gr.,  what  is 
thr  weigh!  of  etch  spoon?  Ans.  loz.  17dwt  ! 

17.    If  ■  family  nee   29gaL  3qt  2gi.  of  molasses  in  G  months, 

what  i>  the  avenge  per  month? 

lqr.  of  hay  is  harvested  from  5  acres,  what 
on  one  acre? 
19.  .  7cwt  2qr.  101b.,  what  is  the 

I  grain-bins  contain  148haek  Spk.  Sqfc  lpt  of  grain, 

wlia;  in? 

21.    It'  a  man   travel   212m.  lfur.  2Grd.   2yd.  in   7  days,  what 

rel  per  da 

v  10. 
15* 


174  COMPOUND    NUMBERS. 

210.  To  divide  by  a  composite  number,  we  may 
divide  by  its  factors,  as  in  division  of  simple  numbers 
(Art.  79). 

23.  Divide  3411b.  Ooz.  12dwt.  by  72. 
lb.  oz.    dwt.       gr. 

9  )  341        0     12        0  First  divide  by  9  and 

8)37     10     14     16  then  Jhe  5U0Jient  b^  8> 

' and  thus  by  72. 

4       8     16     2  0,Ans. 

21.  Divide  396£  2s.  3d.  by 

25.  Divide  725hu>h.  Opk.  6qt.  lpt.  by  81. 

26.  Divide  3971b  llg  75  IB  4gr.  by  63. 

27.  Divide  958m.  5fur.  5ch.  12  li.  5£fin.  by  48. 

211.  When  the  divisor  is  large  and  not  composite,  set 
down  the  work  of  dividing  and  reducing.  There  is  no 
device  for  rendering  the  operation  easier. 

28.  Divide  135bush.  3pk.  3qt  lpt.  by  47. 

bush.     pk.  qt    pt 
4  7)135     3     3     1  (  2bush.  3pk.  4qt.  lpt.,  Ans. 
94 


4  1  bush. 

_4 
,  g  ~    i  Having  found  that  47  is  con- 
14  1       '  tained  twice  in  135,  multiply  47 
by  2,  and  subtract  the  product, 

2  6  pk.  g  },  from  135,  which  leaves  a  re- 

8  maiixlrr  of  41    bushels  ;    reduce 

2  i  i  qt,  the  H   hu-diels  to  pecks,  and  add 

13  3  the  3  peeks,  making  1 67  pecks ; 

— —  then  divide  the  167  pecks  by  47, 

*  ^  *&'  and  so  continue  the  process  till 
the  work  is  done. 

4  7  pt. 

47 


210.    Rale  for  dividing  by  a  composite  number?    211.    Method  of  dividing 
when  the  divisor  is  large  and  not  composite?    Is  there  no  easier  mode? 


MYISIOy.  L75 

29.  If  587  yards  of  cloth  0  4ft.  2d.  l«n\,  what  is  the 
price  per  yard  ? 

30.  Divide  U89gaL  Ipt  8gl  by  ! 

:;i.  A  ftrmer  raited  B5884bush.  8pL  8qt  lpt.  of  corn  on  643 

acresof  land  ;  how  much  was  the  yield  per  acre? 

Suppose  a  man  should  travel  10599m.  ofur.  14r&  4yd  2ft. 

5in.  in  813  days,  what  distance  would  he  travel  per  day? 

33.  in    127  days  a  ship  sals   lis.  9°  4&  8<Fj  what  is  the  dis- 
per  day? 

212.    To  find  the  difference  in  the  longitude  of  two 
places,  when  the  difference  of  time  is  known. 

8  1.  When  it  is  12  o'clock  at  Washington,  it  is  23m.  54gsec. 
past  12  at  Boston ;  what  is  the  difference  in  the  longitude  of 
o  places  ? 

operation.  First   divide   the  23m.  hy 

m.       sec.  4,  became  4m.  of  time  make 

4)23      54J  a  difference  of  1°  of  longi- 

5°    5  8'    3  9"  Am.        tude\  Jhk  &es  5°  an£  a 

remainder  of  3m.     Ihe  3m. 

and  54gsec.  =  234gsec.  The  234£sec.  divided  by  4,  became 
Isec  of  time  make  a  difference  of  1'  of  longitude,  give  58'  and 
a  remainder  of  2$sec.  Finally,  reduce  the  2#sec.  to  60ths  of  a 
ad  divide  by  4,  and  the  quotient  is  39";  i.  e.  the  difference 
in  longitude  between  Boston  and  Washington,  is  5°  58'  39", 
II' nee, 

LB.  Divide  die  difference  in  time,  expressed  in  minutes, 
seconds,  and  OOths  of  a  second,  by  4,  and  the  quotient  is  the  dif- 
ference in  longitude,  expressed  in  degrees,  minutes,  and  seconds. 

Paris  is  2°  20'  15"  east  of  Greenwich  ;  how  many  degrees 
of  Qioenwich  Is  Mew  York,  the  difference  in  time  between 
Paris  and  New  York  being  5h.  om.  21  |sec.  ?    Ans.  74°  0'  3". 

.  i\     The  difference  in  longitude  between  Faris  and  Now  York  is 
_  K  18"  and  this  diminished  by  2'  20'  IS/',  the  east  longi- 
tude of  l'u  :  i    (/  3"  for  the  west  longitude  of  New  York. 

:.    Rule  for  finding  the  difference  in  the  longitude  of  two  places,  when 
the  difference  iu  time  is  knowu.' 


17G  IBS. 

36.  The  difference  in  time  between  Philadelphia  and  Rome 
is  5h.  50m.  30jjsec. ;  Philadelphia  is  75°  9'  west;  what  is  the 
longitude  of  Rome  ?  .  12°  28'  40" 

37.  A  message  telegraphed  from  St.  In  .  29°  48'  i 

at  12  o'clock,  noon,  eras  instantly  received  at  Paris  at  10  h.  10m. 
.  m.,  of  the  same  day  ;  what  is  the  longitude  of  Pari 

38.  At  sun-rise  in  Astoria,  Oregon,  the  bud  is  about  3h.  4l»in. 
6.  high   at   Eastport  in   Maine  ;    what  is  the  difference  in 

longitude  ? 

\   What  is  tin-  difference  in  longitude  between  the  Cape  of 
1    Hope  and  Cape    Horn,  if  a  meteor  seen  at  midnight  at 
Good  Hope  IB  SO  high  a-  to  be  seen  at  the  same  moment  at  ( 
Horn,  the  time  at  Cape   Horn  being   17  minutes  past  6  in  the 
evening?  Ans.  85°  -13'. 

DUODEtrMAI 

213.  Duodecimals  are  compound  numbers  in  which  the 
soal<  uih/  12. 

This  measure  is  usually  applied  to  fact  and  parts  of  a  foot, 
and  i-  used  in  determining  diets  ad  cubic  com* 

Tin-     It-nomination-   an-   feet    (ft.  0C  primes   ('),  secott4I 

("),  thirds  ('"),  fourths  ("  .  ",'",  used  to 

^nominations 

21-1.  The  foot,  being  the  unit,  the  denominations  have  the 
relations  indicated  by  the  following 

TABLE. 
I'    as  ^y     of  a  foot. 

I"     =  TVof   1'     =  y1.Jof      ^     Of  1   ft.  =     T}?      of  a  foot. 

—  ^  of  1"  =  ^  of   T}?  of  1  ft.  =  T7V¥  of  a  foot. 
=  V-T  of  1'"  =  A  of  t^s  of  1  ft.  =  ^v  of  a  foot. 

etc. 

Tims  12  of  any  lower  denomination  make  1  of  the  next 
high' 

"=  1'",  12"'=  1",  12"  =  1',  12r=  1ft. 

313.    What  are  duodecimals?     To  what  applied?     For  what  used?     The  de- 
pominatious?    How  designated?    314.   The  unit,  which  denomination? 


177 

;  r.niACTiON. 

313.    Addition  and  Subtraction  of  duodecimals  are 
irmed  as  the   like   operations   of  other   compound 
numbers. 

I.  Add  together  3ft.  6'  8"  4'"  7"",  9ft.  V  8"  2'"  5"",  and 
4ft.  U'  8"  10"'  8"". 

orEKATiox.  Having  arranged  die  onm- 

9     6'     8"      4"'    7""       ben  as  in  addition  of  com- 

9    7      8        2      5  pound  pombers,  ire  And  the 

1     9      8     10      8  sum  of  the  lowest  denomina- 

Sum,    18     0       1         H~~i  tionto^20-=l'"and8"", 

.*.  set  the  8  in  the  column 
of  fourths,  and  add  the  1  '  to  the  thirds,  and  m  proceed  till  all 
tin-  columns  are  added,  and  so  obtain  18ft.  0'  1"  5'"  8"",  Ans. 

2.  From  6ft.  8'  7"  9'"  3""  take  1ft.  6'  9"  2"'  8"". 

Ai    <S""    cannot    be    taken 

etaauno*.  from  3"",  add  12'"'  to  the  3"", 

Milk,    6     8'       7"     9'"    3""        making   15"",  and    then   take 

Sab,    16        9       2       8  8""  from  the    sum,  giving  a 

•n„_     k     i     7"^      ^      7  remainder  of  7"" ;  then  take 

u,  5     1     10      6       7  .,.„  from  9,„  or  r,  from  ^ 

''  6     8        f      •       I  giving  6'"  by  either   process, 

and  so  proceed. 

3.  Add  10ft.  C  4",  12ft.  9'  8",  and  7ft.  10'  11". 

4.  Subtract  3ft,  8'  4"  3'"  from  9ft.  4'  6"  1'". 

Multiplication. 

216.  Multiplication  of  duodecimals  is  like  multipli- 
cation of  other  compound  numbers,  except  that,  when 
both  factors  arc  in  the  form  of  compound  numbers,  it  is 
required  to  find  tk  nation  of  the  product. 

In  this  investigation, ,/fef  the  tab  of  convenience,  we  familiarly 
speak  of   multiplying  feet  by   feet,  feet  by  inches,   inches  by 

915.  Addition  and  Subtraction,  how  performed?  916.  What  in  Multiplica- 
tion is  peculiar'  Wlint  is  the  multiplier  strictly?  Why  do  we  speak  of  multi- 
plying feet  by  mchct,  etc.  ? 


178  COMPOUND    NUMBERS. 

inches,  etc.,  though  here,  as  everywhere  (Art.  204),  the  multiplier 
rictly  an  abstract  number;  e.  g.,  suppose  a  board  is  10  feet 
long  and  1  foot  wide,  it  evidently  contains  10  square  feet,  and 
if  it  is  10  feet  long  and  2  feet  wide,  it  as  evidently  conta! 

10   square  feet=20  square  feet   (Art.  101),  though  it 
would  be  nom  iirm  that  it  contains  2  feet  timet  10 

still,  wo  are  accustomed  to  say  that  the  area  of  a  board  is  equal 
to  its  length  multiplied  by  i:<  breadth.     Again,  if  a  board  is  10 

!  incfa  wide,  it  contains  ^  as  many  square  fe< 
it   is  feet  in  lew/  itains   ^  of  10   square   feet  = 

;.  ft.  =  1H';  and  if  the  board  is  loft,  long  and  2m.  wide,  it 
contains  -^  of  LOfq.  It.  ss  }|  of  a  sq.  ft.  =  l^aq.  ft-  =  1ft-  and 
8'.     This 

217.    Since  I'-S  ffL,  \"=,  =1  Tl,  ,ft..  ettt,  whether 

the  measure  is  line  .  it  follows  that  1',  in  linear 

meas  >t  in  length  :  i  .  l'is 

an  area,  1  foot  long  and  1    ineh    wide,  end    1"  is  an  area    I    imli 

iqoare;  In  enbfce  meiinre  V  k  ■  $oKd,  l  foot  long,  l  foot  wide, 

and  1  ineh  d< ■•  ,  1  inch  wide, and  1  inch 

a  cubic  inch  ;  etc 

318.     Lot  Of  now  determine  the  denomination  of  the  product 
obtained  by  multiplying  any  two  denominations  together. 


riiiLOSornirALi.Y.                                  iiliarlt. 

2  units  X    3    units    =    6    units,  i.  e.  2ft. 

X3ft. 

=  6ft. 

1     M     X   f\  unit     =  ft  unit,    i.  e.  2ft. 

X3' 

=  6* 

2     «     XTf*    "        =rf*    "      >-e.2ft. 

X3" 

=  6" 

etc. 

etc 

-j^unitX     tV     unit=    r|T    unit,  i.  e.  2' 

X3' 

=  6" 

A    u    X    X             =  TVW     "     i.  e.  2' 

X3" 

Estf" 

rV    "    X  TAi     "    =nhi    "     i-e-2 

X3'" 

=  6 

etc 

etc. 

T^-unitX     Tf*    «nit=    ^ft^    unit,  i.  c. 

r  X  3" 

==  G"" 

tIt      "      X    T?V?      "     =    T¥^ffTJ                L  e- 

2"  X  3'" 

__  6///// 

lU    "    XW»W    "    =mfm    "     i-e. 

2"  X  3"" 

' nilll/t 

etc 

etc. 

«517.    What  U  1'  in  linear  measure ?     1'  in   square   measure?    1"  in  tquara 
■M««l»f    V  in  cubic  meagre?    1"?    Y"1    1""? 


what  . 

9 

OPERA 

6 

7 

9" 

2 

r 

5" 

13 

3' 

6" 

3 

10' 

a" 

3'" 

2' 

9" 

2'" 

9"" 

IT.' 

II  ince,  to  determine  the  denominmtioB  of  the  product 
of  tu  in  duodeoim 

BULB.     Add  the   indict*  of  factors  together,  and  the 

t     ndex  of  the  product. 

in  length  n  •","  b  breadth] 

First,  9"  X  2=18" 
=  1'  6";  the  6"  we  write 
UOder  the  seconds,  and 
reserve  the  1'  to  add  to 
the  next  product,  thus, 
7'  x  2=lf.  which  Su- 
ited by  the  1/  previ- 
Ans.   17       4'    9"    ft'"    9""       onely  obtained  gtae  W 

=  lft.  3';  the  §  Ifl  writ- 
•wn,  and  the  1ft.  is   carried  to  the  product  of  the  feet,  mak- 

[n  like  manner  we  multiply  by  the  V  and  then  by 

the  partial   products  as   in   the   margin*      Finally, 

•  partial  product!  i-  the  product  aought     II 

219.    To  perform  Multiplication  of  Duodecimals, 

Kii.k.      Ihj  the  ride  for  multiplication  of  compound  numbers, 

multiply  the  multiplicand  by  each  term  in  the  multiplier,  and  write 

■■/is  if  the  second  partial  products  in  the  order  of  their  values, 

so  that  similar  terms  shall  stand  in  a  column  together  ;  the  sum  of 

the  partial  products  will  be  the  entire  product. 

2.  3. 

Multiplicand,  8     4'     G"  4        8'     9" 

Multiplier,       2     8'     5"  2        3'     7" 

G     9'     0"  9        5'     6" 

2     3'     0"      0'"  1        2'     2"    3"' 

1'     4"    1  ()'"    G""  2'     9"    1"'    3"" 


9      1'     4"    10'"    6""      10     10'     5"   4"'   3'"' 

1.  What  quantity  of  board*   will  be  required  to  lay  a  floor 
long  end  .    103ft.  10'  6"  2'". 

218.    Rule   for  determining  the  denomination  of  ft  produ.  philo- 

sophically and  familiarly.    210.  Rula  for  multiplication  of  duodecimals? 


180  COMPOUND    .\ 

0.  What  are  the  contents  of  ■  granite  block  that  is  Cft.  3'  long, 
2ft.  4' wide,  and  lft.  3'  thick? 

Ans.    18ft.  2y  9".     (See  Art.  104). 

G.  How  many  feet  of  flag-stone  in  a  walk  loft.  0'  long  and 
3ft.  4'  wide? 

7.  How  many  solid  feet  of  marble  in  a  block  that  is  8ft.  3' 
long,  3ft.  6'  wide,  and  lft.  4'  thick  ? 

8.  How  many  cubic  feet  of  earth  must  be  removed  in  digging 
a  cellar  15ft.  6'  long,  12fu  8'  wide,  and  6ft.  8'  deep? 

9.  How  many  feet  in  a  stock  of  8  boards,  that  are  10ft.  8' 
long  and  10'  wide  ?  Ans.  71ft.  1'  4". 

10.  How  many  feet  of  boards  1'  thick  can  be  sawed  from  a 
a  stick  of  timber  that  is  12fu  8'  long,  10'  wide,  and  8'  4"  thick, 
provided  no  timber  is  destroyed  by  the  saw-cut? 

11.  How  many  cords  of  wood  in  a  pile  that  is  18ft.  6'  long, 
6ft.  8'  high,  and  4ft.  wide? 

12.  How  many  square  yards  of  carpeting  will  cover  a  room 
that  is  18ft.  long  and  16ft.  6'  wide? 

13.  Multiply  3ft.  6'  4"  by  8ft.  9'  6". 

Division. 

2*20.  Division  of  duodecimals  is  like  division  of  other 
compound  numbers. 

Ex.  1*.  Divide  24ft.  10'  10"  4'"  by  7.     Also  by  9. 

OPERATION.  OPERATION. 

7  )  2  4     1  0'     1  0"     4'"     9  )  2  4     10'     1  0"     4" 

~2        lP        2"    5"    9""     4""' 

3. 
6)45     4'     1"     6"' 

Note.  When  both  dividend  and  divi.sor  are  expressed  as  compound 
numbers,  they  m:iy  be  reduced  to  the  smallest  denomination  in  either;  after 
which  divide,  and  the  quotient  will  be  units,  i.  e.feet;  thus,  68ft.  l(y  8" 
divided  by  2ft.  8'  equals  9920" V  384"  =  25 Jf,  i.  e.  25ft.  10',  Ans. 

320.  How  ia  division  of  duodecimals  performed?  How  when  the  divisor  is 
compound  1 


ts.  3 

6' 

2. 

8" 

4"' 

8) 

31 

6' 

8" 

8'" 

misci:i.i.\  181 

4.  The  area  of  a  floor  is  197ft.  V  8",  and  the  length  of  the 
floor  is  1.0ft.  8';  what  ii  its  width?  12ft.  7'. 

5.  The  area  of  a  garden  walk  is  89ft.  4'  and  its  width  is  2ft.  8'* 
what  is  its  length? 

Miscellaneous  ExAMFLM  n  RD  Ximhers. 

1.  If  I52bush.  '"'pk.  3<[t.  Ipt.  of  wheat  grow  on  9  acres  of 
land,  how  many  bushels  grow  on  7  acr< 

2.  A  man  having  207m.  4fur.  25rd.  1yd.  to  travel  in  6  days, 
80m.  8fur.  2  "ml.  5yd.  on  die  first  day,  and  33ra.  4fur.  20rd. 

4yd.  on  the  second  day;  how  far  per  day  must  he  travel  to  finish 
the  journey  in  the  remaining  4  da; 

3.  Multiply  3£  15s.  Gd.  lqr.  by  857,  and  divide  the  product 
by  167. 

4.  I  have  a  stock  of  9  boards,  which  are  12ft.  8'  long  and  10r 
wide.  With  these  boards  I  wish  to  lay  a  floor  15ft.  in  length ; 
how  l  I  make  it  ? 

5.  If  1  cubic  foot  of  water  weighs  02 lb.  8oz.,  and  if  a  cubic 
foot  of  granite  weighs  2L  times  as  much,  what  is  the  weight  of  a 
block  of  granite   12ft.  long,  lft.  s'  wide,  and  9'  thick? 

6.  From  the  sum  of  3\vk.  6d.  lOh.  20m.  18sec.  and  2wk.  3d. 
50m.  aOsec.  take  the  difference  between  Cwk.  5d.  8h.  25m. 

30sec.  and  5wk.  2d.  22h.  18m.  15 

7.  What  ifl  the  difference  in  time  between  Amsterdam  4°  44' 
oast  longitude,  and  Annapolis  7G°  43'  west  longitude? 

8.  When  it  is  noon  in  Dublin,  G°  7'  13"  west  longitude,  it  is 
10m.  and  16}|SCC.  past  8  o'clock  in  the  evening  in  Peking; 
what  is  the  longitude  of  Peking? 

i  tow  many  days,  hoars,  eta,  from  30m.  20sec.  past  3  o'clock. 
i\  m..  Peh.  8,  1864,  to  40m.  toseo.  past  8  o'clock,  a.  m.,  July  4, 
.  reckoning  each  month  at  its  actual  length? 
LO.  Bought  8cwt  8qr.  181b.  of  sugar  at  8£c.  per  pound,  and 
sold  £  of  it  at   8c.  and  the   remainder  at   Die.  per  pound;  what 
gained  by  the  transaction 
11.  What   ifl  the  value  in  Avoirdupois  Weight  of  241b.  6oz. 
llMv,  :,t? 

16 


182  COMPOUND    NUMB! 

12.  How  long  a  time  will  be  required  for  one  of  the  heavenly 
bodies  to  move  through  a  quadrant  of  a  circle,  if  it  moves  at  the 
rate  of  V  3"  per  minute  ? 

1.;.  Th«    distance   from  Eastport,  Maine,   to  San   Fran< 
California,  is  about  27G0  miles.      If  a  man,  starting  from   East* 
port,  travel  toward   San    Francisco  for  75  days,  at   tin-  rate  of 
iMm.  3fur.   20rd.   per  day,  how  far  will  he  then  be  from 
Francisco  ? 

1  1.  A  certain  island  il  75  miles  in  circumference.     A  and  B, 
start  tag  at  the  same  time,  and  from  the  same  point,  and  going  in 
tli--   same  direction,   travel    round  this   bland,  A  at  the  rate  of 
24m.  3fur.  lOrd.,  and  B  at  the  rate  of  L5m.  C.fur.  20rd.  per  d 
how  tar  apart  air  A  and   15  at  the  end  of  live  days? 

15.  A  merchant  bought  L  25  barrels  of  floor,  at  IX  15s.  6cL 

per  barrel,  and  afterward  exchanged  the  flour  for  260  yards  of 
broadcloth,  which  he  sold  at  18s.  9d.  3qr.  per  yard;  did  he 
gain  or  lose,  and  how  much  ? 

16.  How  many  feet  of  boards  will  be  required  to  make  12 
boxes  whose  interior  dimensions  are  5ft.  6',  4ft.  9',  and  3ft.  8', 
the  boards  being  1'  in  tliicki: 

17.  I  low  many  feel  less  are  required  to  make  12  boxes  whose 
exterior  flimmikmi  are  like  the  interior  of  those  in  Ex.  16,  the 
boards  being  of  the  same  thickness?  Ana.  111ft.  4'. 

18.  Wnal  is  the  difference  of  the  capacities  of  the  two  sets  of 
boxe  !  in  I.\.  16  and  17  !  Ans.  122ft.  10'. 

19.  How  many  times  will  a  wheel  Oft.  8in.  in  circumference 
turn  round  in  running  from  Boston  to  Worcester,  a  distance  of 
•1  lm.  4 fur.? 

20.  How  many  gallons,  wine  measure,  in  a  water  tank  4ft. 
Gin.  long,  oft.  8in.  wide,  and  3ft.  9in.  deep  ? 

21.  If  a  teacher  devote  5h.  30m.  per  day  to  50  pupils,  what 
i<  thi  average  time  for  each  pupil  ? 

22.  If  a  man,  employed  in  counting  money  from  a  heap,  count 
75  -ilver  dollars  each  minute,  and  continue  at  the  work  12  hours 
each  day,  in  how  many  days  will  he  count  a  million  of  dollars? 

23.  How  many  pounds  of  iron  in  one  scale  of  a  balance,  will 
poisa  75  pounds  of  gold  in  the  other  6cale  ? 


183 


DAG  i:. 

*>*J1.     |  i.  ||   ■  contraction   of  per  cen turn,  a  Latin 

phrase  which  meant  by  the  kttndnd;  thai)  ten  per  cent,  of  a 
boshd  of  corn  means  tea  one-hondredthfl  of  it  ;  i.  <•.  tea  parts 

:v  hundred  pa  per  cent,  of  a  sum  of  m 

I  onc-hundredths  of  the  sum,  i.e.  $6  out  of  every  $100  ; 

\*..i  I  of  the  words  p^r  cent.,  it  is   eastOnMOyto  use  this  sign, 

°/  ;  t!.  it.  is  written  b°/g  ;  \\  i>cr  cent.,  ±\°/0- 

5jtJ%J.  The  RATE  vvk  I  IEHT.  IS  the  number  for  each  hundred  ; 
(has,  6J£  is  ^...or  .06,  i.  e.  G  parts  for  each  hondred  \ 

£££3.    The  Pi  <B<  il  the  sum  computed  on  the  given 

numix-r;  thotj  ili»'  percentage  on  -^200  at  6  per  cent,  i- 

2*2  I.    The  Base  of  percentage  ■  the  number  on  which  the 
is  computed  ;  thus,   S200  is  the  base  on  which  the 
competed  ia  Art  223;  a  bushel  of  corn  is  the 
ant  bate  mentioned  ia  Art.  221. 

*2"2«I.  The  rate  per  cent.,  being  a  certain  number  of  hun- 
dredths, may  be  expressed  either  decimally,  or  by  a  common 
fraction,  as  in  the  following 


1 
2 
5 
6 
10 

100 
125 


per  cent, 
per  cent. 

per  cent. 
per  cent. 

G^  p< •:• 

8£  per  cent. 
12£  per  cent. 


IS 


TABLE. 

Decimals. 
.01 

.10 

.50 
1.00 

•08^ 
.125 


Common  Fractions. 


eto. 


Ttlo" 

1  |  .7 

r\ft 

m 

m 

TTJff 

«tc 


1. 
1*. 


184  PER< 

Note.  When  the  per  cent,  is  expressed  by  a  decimal  of  more  than  2 
places,  the  figures  after  the  second  decimal  place  must  be  regarded  as  parts 
of  1  per  cent. ;  thus,  (in  the  last  line  of  the  foregoing  table,)  .125  is  12^ 
or  12^  per  cent. 

1.  Write  the  decimal  for  4  per  cent.  Ans.  .04. 

2.  Write  the  decimal  for  8  per  cent. ;  12  per  cent. ;  1G£  per 
cent. ;  25  per  cent.;  72  per  cent. 

3.  Write  the  common  fraction  for  1C§  per  cent. ;  20  per  cent. ; 
33£  per  cent.;  7.3  per  cent.  1st  Ans.  |. 

Problem   1. 

220.  To  find  the  percentage,  the  base  and  rate  per 
cent,  being  given. 

Ex.  1.  B  had  $175,  but  lost  8  per  cent  of  it;  how  many 
dollars  did  he  lose? 

$  1  7  5  Since  8  per  cent,  is  .08  =  f5,  the 

.0  8  M  is  found  by  multiplying  $175   by 

$TT00,   A  .08  or  by  ^.     Hence, 

BULK  1.  Multiply  the  base  by  the  per  cent.,  written  decimally ;  or, 

Kiii  J.  Find  such  part  of  (he  base  as  the  rate  is  of  100 
(Art.  IS1), 

2.  A  farmer  having  48  sheep,  lost  25  per  cent  of  them  ;  how 
many  did  he  lose  ? 

By  Rule  2. 

\  of  48  =  12,  Ans. 
Or,    48  X  i  =  12,  Ans. 


By  Rule  1 
48 
.2  5 

' 

240 
96 

1  2.0  0, 

Ans. 

221.  Meaning  of  per  cent.?  222.  Hate  per  cent.?  223.  Percentage? 
224.  Base  of  percentage?  225.  In  what  ways  may  the  rate  be  expressed? 
If  expressed  decimally  by  more  than  two  figures,  what  are  the  figures  after  the 
second  decimal  place?  226.  Rule  for  finding  percentage  when  the  base  and 
rat*  are  given?     Second  Rule? 


185 

8.  What  is  6  per  cent,  i  $15. 

4.  What  is  8  percent  of  H 

5.  WhaJ  rent,  of  $600? 

ft,  What  is  8J  SOOboih.  of  wl, 

i-h. 

7.  What  r  cent,  of  12001b.  of 

8.  A  farm.-r  cultivate-  85  aero  of  corn  this  year,  and  into 

to  cultivate  20  per  cent,  more  next  year;  how  many  acres  does 
be  intend  to  cultivate  next  year?  Au- 

la an  Orchard  of  900  trees,  83£  per  cent,  are  peach  t, 

how  many  peach  trees  are  there  in  the  orchard? 

L0,  A  teacher  pronounced  56  words  for  his  papOa  to  spell, 
but  L4f  per  cent,  were  mil  Bpnllfid ;  how  many  words  were  mis- 

11.  Only  86|  per  cent,  of  a  class  of  27  pupils  solved  a  problem 
given  them  Oq  ;   how  many  of  the  claSS  failed? 

ij.  The  population  of  a  certain  city  is  18775,  what  will  it  be 
m  one  year  from  this  time  if  it  gains  8  per  cent.? 

18.  The  population  of  a  certain  State  is  1876875,  what  will  it 

be  in  one  year  if  it  loses  12  per  cent.? 

1 1.  A  and  B  oosninenced  business,  each  with  $8456.    A  gained 

26  per  cent  and  li  i  .  ;  how  much  was  A  then  worth 

more  than  B? 

l."».   A  speculator  paid  r  a  lot  of  flour,  and  lost  9  per 

cent.;  for  what  sum  did  he  sell  the  flour? 

16.  One  acre  of  corn  y it  Ids  80  bushels,  and  another  acre  20 
:  how  many  bushels  does  the  second  acre  yield? 

Pboblu  -• 

227.    To  find  the  rate  per  cent  when  the  base  and 
i  are  giren. 

hne  dollar  is  what  per  cent,  of  $1? 

.j  j  |  One  dollar  is  \  of  Si.  and  }  reduced 

decimal   IS  .25  ;   i.  • 

2o>  oent  ot'sj.    Tlie  hubm  raeutl 

by  multiplying  $1  by  100,  and  dividing  the  prodnd  by  1.    1 ; 
1G* 


186  PERCENTAGE. 

Rule.  Multiply  the  'percentage  by  100,  and  divide  the  product 
by  the  base. 

Note.  This  rule  is  the  converse  of  that  in  Art.  226  ;  thus,  25  per  cent, 
of  S4  is  $4  X  .25  =  $1 ;  and,  conversely,  $1.00  ~  $4  =  .25,  i.  e.  l'j  per  cent. 

2.  What  per  cent  of  $150  is  $18? 

1800  -7-  150  =  12  per  cent.,  Ans. 

3.  What  per  cent  of  $300  is  $19  ?  Ans.  6$  per  cent. 
-1.  AVliat  per  cent,  of  S350  is  $43.75?     Ans.  12£  per  cent 

5.  What  per  cmt.  of  $340  h  $84  ? 
ft.  What  per  cent  of  $64  is  $16? 

7.  What  pet  cent,  of  $1000  is  $5?       Ans.  £  of  1  per  cent 

8.  B  inherited  $3500,  and  in  8  months  spent  $870  ;  what  per 
cent,  of  his  inheritance  did  he  spend?  What  per  cent,  had  he 
remaining?         Ans.  Spent  25  per  cent.,  and  had  75  per  cent. 

9.  Outofaca.sk  of  wine  containing  96  gallons,  32  gallons  were 
Irawn;  what  per  cent,  of  the  whole  remained  in  the  ca>k  ? 

10.  A  merchant  having  $1000,  deposited  $650  in  a  bank; 
What  per  cent  of  his  money  did  he  depoafl  ? 

11.  A.  teacher  baring  a  salary  of  $2400,  spends  $2000  an- 
nually ;  what  per  cent,  of  h,  does  he  sa\ 

Pn<  • 

228.  To  find  the  base  when  the  percentage  and  the 
rate  are  given. 

Ex.  1.   |6  is  3  per  cent  of  what  sntn  ? 

'•  is  3  per  cent,  then  1  per  i  of  $6,  which  is  $2, 

and  if  $2  is  1  per  cent.,  then  100  per  cent  i-  LOO  times  $2,  which 

0;.«.  si*)  i>  ">  per  eent  of  $200,  Ana, 

The  same  result  is  obtained  by  first  multiplying  SC  by  100,  and 
then  dividing  the  product  by  3  ;  thus,  $600  -r-  3  =  $200,  Ans. 
Hence, 

Bui  B.  Multiply  the  percentage  by  1 00,  and  divide  the  product 
by  the  rate. 

227.  Rule  for  finding  the  rate  when  the  base  and  percentage  are  known? 
What  of  this  rule,  and  that  in  Art.  226?  238.  liule  for  finding  the  base  when 
the  percentage  and  rate  are  known? 


2.  $0  ia  l  per  cent,  of  what  .sum?  Aim 

3.  $  per  crnt.  of  "hat  sum?  160. 

4.  $12  ia  7  per  cent  of  what  sum?  7i.l2f. 
<>  i^  16  per  cent  of  what  mm? 

6.  12  m  8  per  cent  of  what  numl  •  400. 

:.  of  what  number? 

8.  33  is  1|  per  cent,  of  what  number? 

9.  A  fanner  bought  a  form  tor  $27.~>C>,  which  was  25  per  cent: 
of  his  property;  what  waa  his  property?  An-.  $11024. 

10.  A  man  sold  56*  geese,  which  was  28  per  cent,  of  his  flock  ; 
how  many  geeM  had  he? 

11.  A  merchant  having  a  quantity  of  flour,  bought  600  barrela 

more,  when  he  found  that  the  quantity  bought  was  75  per  cent 

of  all  lie  then  had  ;  how  many  barrels  had  he  before  he  bought 
the  last  lot?  Ans.  200. 

19.  r  saves  $400  annually,  which  is  16§  per  cent,  of 

ury  ;  what  h  his  salary? 

13.  The  population  of  a  town  was  7G9  greater  in  1800  than 
in  1850,  and  this  was  an  increase  of  20  per  cent,  on  the  popula- 
tion of  1850 ;  what  was  the  population  in  1850  ? 

I  XT K  REST. 

229.     Interest  is  money  paid  for  the  use  of  money. 
Tin-  Pkin(  ii-al  is  the  sum  for  which  interest  is  paid. 
Am ui  NT  is  the  sum  of  the  principal  and  interest. 
3ttO.     An  example  in  interest  is  only  a  question  in  percentage. 
The  principal  i>  th»-  bate  of  percentage  (Art.  224  ).  the  interest  is 
\agt  (Art.  228), and  the  interest  on  SI  for  a  year  is  the 
i  ritten  fcchnaHj  (  A 
£31.    The  rate  i-  usually  %/7.#W  By  far,  and  a  higher  rate  than 

the  law  all  n/. 

In  and  and  most  of  the  United   States  the  legal  or 

rial   the    Principal'     Amount?      230.    I. 
what  relation  to  percentage?     What  is  the  base7     The  percentage?     The  rate* 
rate  fixed  I     What  is  usury  ?     Name  tin-  legal  rate  in  6ome 

: 


188  PERc 

lawful  rate  is  6  per  cent. ;  in  New  York,  7  per  cent. ;  in  most  of 
the  Western  States,  M  high  SI  10  per  cent,  by  agreement;  in 
Texts,  as  high  as  12  per  cent  f  in  California,  any  rate  by  agree- 
ment, etc.  On  debts  In  favor  of  the  United  States,  G  per  cent. 
In  France  and  England,  5  per  cent. 

Note.    In  this  treatise,  6  per  cent,  is  understood  when  no  per  cent,  is 

mentioned. 

232.  When  the  rate  is  6  per  cent.,  the  interest  of  $1  for  a 
is  6c.  j  for  2  yea i ft,  1_< •.,  etc. ;  for  1  month,  fa  of  Gc.  =  5  mills 
or  ^c. ;  for  2  months,  lc. ;  3  months,  l£c. ;  G  months,  3c.;  9 
months,  4^c,  etc  ;  for  1  day,  ^  of  5  mills  =  £  mill ;  for  2  d 
$m. ;  3  days,  £ra.;  4  days,  §m. ;  6  days,  gin.;  6  days  lm. ; 
7  days,  l.Wn.;  9  days,  l$m. ;  12  days,  2m.;  24  days,  4m.;  etc., 
etc     Hence, 

To  find  the  interest  of  81  at  6  per  cent,  for  any  time, 

BULB.  Tak$  Gc.  (=  $.0G)  for  each  year,  lc.for  each  2  months 
in  the  part  of  a  year,  5  mills  (=  $.005)  for  the  odd  month, 
if  there  be  one,  and  1  mill  for  each  6  days  in  the  part  of  a 
mojith. 

1.   What  is  the  interest  of  $1  for  3yr.  9m.  18d.? 

OPERATI 

$.1  8     =  interest  of  $1  for     3  years 

.0  4  5  =       "  u      9  months 
.0  0  3  =       "        "     "     "   1_8  da 

$.2  2  8=       "         "     "     "      3yr.  9m.  18d.,  Ans. 

2.  Wbti  is  the  interest  of  $1  for  2yr.  5m.  20d.? 

OPERATION. 

I        =  interest  of  $1  for     2  years. 
.0  2  5     =       "         "     "     "      5  months. 
.003$  =       "         «     "      "20  days. 

$.1  4  8  £  =       «        «     «     "      2yr.  5m.  20d.,  Ans. 

5131.  What  will  be  understood  when  no  rate  is  mentioned  ?  233.  Kule 
for  finding  the  interest  of  SI  at  6  per  cent  for  any  given  time? 


in  189 

With  very  little  practice  the  ptij.il  will,  without  making  n  | 
mentally  determine  the  i-  -iv  length  of  time.    This  hahit  is 

very  desirable,  as  it  will  ( 

\  hat  ii  the  interest  of  Si  for  8jrr.  lm.  I 

Ans.  $.187$. 
1  What  is  the  interest  of  $1  for  lyr.  3m.  I 

Ans  $.079g. 
of  Si  for  4yr.  2m.  ' 

Ans.  $.250§. 
G.  What  is  the  intm-t  of  $1  for  4yr.  5m.  17.1.? 

7.  What  is  the  interest  of  $1  for  4yr.  9m.  IlM.? 

8.  What  ifl  the  interest  of  $1  for  lOyr,  11a,  7d.? 

9.  What  ifl  the  interest  of  $1  for  2yr.  11  in. 
10.  What  is  the  interest  of  SI  for  lyr.  8m.  3d.? 

233.  To  find  the  interest  of  any  sum  at  6  per  cent, 
for  any  given  time. 

The  interest  of  $2  is  evidently  twice  as  much  as  the  interest 
of  $1  ;  so  the  interest  of  $3,  $4,  or  $7,  is  3,  4,  or  7  times  (he 
Bt  of  SI  ;  and  the  interest  of  S2.25  is  2.25  (i.  e.  2  and  26 
hundredths)  times  the  interest  of  $1  ;  .*.  to  find  the  interest  of 
any  number  of  dollars  we  have  only  to  find  the  interest  of  $1, 
and  then  multiply  the  interest  by  the  number  of  dollars  in  the 
princijuil. 

11.  What  ifl  the  interest  of  $2  for  lyr.  5m.  9d.? 

S  .0  8  6  $  =  interest  of  Si  for  lyr.  5m.  9d. 
2 

$.173  =  interest  of  S2  for  lyr.  5m.  9d.,  Ans. 

12.  What  is  the  interest  of  S6.50  for  3yr.  8m.  18d.? 

=  interest  of  SI  for  3yr.  8m.  18d. 


11150 
1338 


$  1.4  4  9  5  0  =  int  -G.50  for  3yr.  8m.  18d.,  Ans. 

»33.  What  is  the  Note  ?    233.    Rule  for  finding  the  interest  of  any  sum  at 
■  nt.,  for  auy  time'     IU-asou? 


190 


1'KIK 


13.  What  is  the  interest  of  $300  for  2yr.  7m.  2 1.1.  ? 

$.159  =  interest  of  $1  for  2yr.  7m.  24<L 
300 

$  4  7.7  0  0  =  interest  of  $300  for  2yr.  7m.  24d.,  Ans. 

14.  What  is  the  interest  of  $700  for  lyr.  9m.  12d.  ? 

Ans.  $74.00. 

15.  What  is  the  interest  of  $400  for  2yr.  6m.  16d  ? 

16.  Wliat  is  the  interest  of  $350  for  3yr.  8m.  24d.  ? 

234.  The  mode  of  casting  interest  given  in  Art.  233  is  per- 
fectly simple,  but  the  product  is  not  changed  when  the  multipli- 
cand and  multiplier  change  places  (Art.  59,  Note).     Hence, 

To  cast  interest  at  6  per  cent,  per  annum,  on  any  sum, 
for  any  time, 

Rule.  Multiply  the  principal  by  the  decimal  which  represents 
the  interest  of$l  for  Qie  given  time. 

17.  What  is  the  interest  of  $468  for  2yr.  6m.  lid.? 

FI^ST   OPERATION. 

#  =  ^  -f-  J.  Instead  of 
multiplying  by  £,  as  in  this 
example,  it  is  usually  easier 
to  multiply  by  ^  and  £,  i.  e. 
divide  by  2  and  3,  as  in  the 
following  operation : 


$4  6  8.    =  Principal. 
.1  5  1  ^  =Int.  of  $1. 

890 
468 

2340 

468 

$7  1,0  5  8,  Ans. 

SECOND    OPERATION. 
$4  6  8. 

1.5  H* 

234 

156 
468 

2340 

468 

$7  1,0  5  8,  Ans. 

In  like  manner,  when  the 
multiplier  is  §,  we  may  divide 
by  3  and  set  down  the  quo- 
tient twice. 


334.    Second  rule?     Beason?     ftiriett  way  of  multiplying  by   five  sixthtf 
Why  correct?     Easiest  way  for  two  thirds? 


INTEREST.  191 

.   What  i|  til.-  $48.50  for  2yr.  7ra.  21d.  ? 

Ans.  $7.6872:». 

19.  What  is  the  interest  of  $248  for  2yr.  8m.  18d.? 

Ans.  $84£24 

20.  What  is  the  interest  of  $965,188  for  2yr.  8m.  1  Ld.? 

7      . 

PBl.     In  the  20th  example  the  Ans.  is  S132.06989H,  but  this,  in  all 
tran-u.  turns,  would   bo  call<  In  the  following 

<-\;imj)li'>  in  interest   only  .'J  decimal   places  in  the  product  will   DC  pre 

but  if  the  4th  decimal  place  is  5  or  more,  the  third  place  will  be  increased  l»y 
indth. 

21.  What  is  the  interest  of  $225.87  for  lyr.  ftm.  1  -"„!.  ? 

Ans.  $17,505. 
What  is  the  interest  of  $85.40  for  2yr.  6m.  9d.? 

Ans.  $5,363. 

23.  What  is  the  interest  of  S  150.87  for  lyr.  7m.  9d.? 

24.  What  is  the  interest  of  $375.50  for  2yr.  lm.  8d.? 
What  is  the  interest  of  $225.75  for  lyr.  5m.  12d.? 

26.  What  is  the  interest  of  $84.82  for  2yr.  4m.  18d.? 

27.  What  is  the  interest  of  $125.16  for  lyr.  11m.  25d.? 

28.  What  is  the  interest  of  $658.25  for  lyr.  2m.  13d.  ? 

29.  What  is  the  interest  of  $125  from  June  7,  1851,  to  Feb. 
11,  1^  Ans.  $20,083. 

Note  2.     Ex.  29  differs  from  the  preceding  only  in  its  being  necessary  to 
find  the  time  (Art.  203). 

30.  Find  the  interest  of  $154.25  from  April  18,  1852,  to  Jan. 
26,  1855.  Ans.  $25,657. 

31.  Find  the  interest  of  $172  from  Aug.  7,  1854,  to  Sept.  9, 
1856. 

32.  Find  the  interest  of  $254  from  Nov.  12, 1855,  to  Jan.  30, 

What  is  tin-  interest  of  $132.25  from  Nov.  13,  1836,  to 
j.  L841? 
34.   What   is    thfl   int.  rest  of  $100   from  March  26,  1841,  to 

*34.    What  of  decimal  placet  after  the  third  in  the  Ant.t    Explain  Ex.  29. 


235.  To  find  the  interest  when  the  principal  is  in 
pounds,  shillings,  pence,  and  farthing 

BULK.     Reduce  the  lower  denominations  to  the  decimal  of  a 
pound  (Art.  175),  then  proceed  at  icith   dollars  and  cents,  and 
!fy  reduce  the  decimal  part  of  the  interest  back  to  shillings, 
pence,  and  farthings  (Art.  176). 

'it  3  decimal  places  in  the  multiplicand  are  used. 

35.  What  is  the  interest  of  56£  10s.  6d.  3qr.  for  1   yr.    6m. 

Ans.  5£  6s.  3d.  lqr. 

36.  What  is  the  interest  of  246£  18s.  Od.  lqr.  for  2yr.  3m. 
15d.? 

37.  W.  rest  of  125£  16s.  8d.  2qr.  from  Nov.  13, 
1861,  to  March  26,  1863? 

236.  To  find  the  interest  of  any  sum  for  any  time,  at 
any  other  rate  than  6  per  cent.: 

I  { r  i .  r. .  /  V rsf  find  the  interest  at  6  per  cent. ;  then  divide  th  is 
interest  by  <*,.  wiM  WtBfim  tk$  interest  at  1  per  cent.;  and, 
finally,  multiply  the  interest  at  1  per  cent,  by  the  given  rate. 

88.  What  is  the  interest  of  $124.50  for  lyr.  4m.  12d.,  at  5 
per  cent.? 

OPERATION. 

$  1  2  4.5  0,  Principal. 

.0  8  2  =  Int.  of  $1  at  6  per  cent,  for  lyr.  4m.  12d. 

24900 
99600 

6  )  $10.20900  =  Int.  of  Principal  at  G  per  cent. 

$  1.7  0  1  5     =  Int.  of  Principal  at  1  per  cent. 


$  8.5  0  7  5      =  Int.  of  Principal  at  5  per  cent..  An-. 
39.  What  is  the  interest  of  $9  t   lyr.  9m.   18d.,  at  8 

percent.?  Ans.  $49,284. 

235.  Rule  for  casting  interest  on  pounds,  shillings,  etc.  ?  How  many  decimal 
places  in  the  multiplicand  are  used?  236.  Rule  for  computing  interest  at  any 
given  : 


•r. 

40.  What  ia  the  interest  of  $256.84  for  lyr.  3m.   15d.,  at  9 
per  cent.  ? 

41.  What  is  the  interest  of  24£  6s.  8d.  lqr.  for  2yr.  9ra.  12d\, 
at  5  per  cent  ?  84  8qr. 

48.   Wl,a:  u  the  interest  of  150£  10s.  for  2yr.  4m.  6d.,  at  4$ 
nt.  ? 

237.    To  find  the  amount  of  any  sum  at  any  rate  for 

any  ti. 

'  find  the  interest  by  the  preceding  rules,  and  to 
the  interest  add  the  principal. 

43.  What  is  the  amount  of  $325.75  for  lyr.  4m.  24d.,  at  6 
per  cent.  ? 

OPERATION. 

$  3  2  5.7  5,  Principal. 

.084  =  Int.  of  $1  for  lyr.  4m.  24<L 

130300 
260600 


S  2  7.3  6  3  0  0  =  Int.  of  Principal. 
$  3  2  5.7  5  =  Principal. 

$  3  5  3.1  1  3       =  Amount,  Ans. 

44.  What  is  the  amount  of  $224.48  for  2yr.  6m.  15d.? 

Ans.  $258,713. 

45.  What  is  the  amount  of  $48.33  for  lyr.  6m.  ? 

46.  What  is  the  amount  of  $365.25  for  lyr.  3m.  9d.? 

47.  Wrhat  is  the  amount  of  $444  from  July  18,  1861,  to  Sept. 
4,  1862?  Ans.  474.044. 

48.  What  is  the  amount  of  $32.25  from  Nov.  15,  1860,  to 
July  25,  1862,  at  7^  per  cent.  ? 

49.  What  is  the  amount  of  $187.44  from  May  25,  1859,  to 
April  19,  1861,  at  7t\  per  cent.  ? 

50.  What  is  the  amount  of  82£  12s.  6d.  3qr.  from  Feb.  12. 
1860,  to  Dec  24,  1862,  at  5  per  cent.? 

Rule  for  finding  the  amount  of  any  sum  for  a  given  time  and  :»te* 
238.     Rule  for   casting  interest  oa  notes  when   partial    payments  have  bo«a 
•dn  Ex.51. 

17 


II 4  PEBOXNl 

238.    To  cast  interest  on  Notes  when   Partial  Pay- 
nicnts  hare  been  made: 

K;  LB.      Find  the  AMOUNT  of  the  principal  to  the  time  of  the 
first  payment  ;  from  this  amount  subtract  the  first  payment,  and 
the  ici.m  aim  .Kit  is  a  NEW  principal,  with  which  proceed  to  the 
of  the  second  pay  I  so  on  to  the  time  of  settlement. 

Exception.     If  any  payment  is  less  Otan  Hie  interest  due, 
mst    the   interest  on  the  same   principal    up  to  the  frst 
tclien  the  sum  of  the  payments  shall  equal  or  exceed  the  interest 
aue  ,  then  subtract  the  SUM  of  the  payments  from  the  A.MO 
of  the  principal,  and  the  remainder  is  a  new  principal,  with 
whirh  proceed  as  before. 

51.  $525.  Andover,  Mass.,  June  4,  1848. 

For  value  received,  I  promise  to  pay  John  Davis,  or  order, 
five  hundred  and  twenty-five  dollars,  on  demand,  with  ii 

Da  mi  i.   Ti:i 

On  this  note  are  me  following  ind  Sept.  9,  1849, 

$114.20;   May  15,   1850,  $78,285;  Aug.  6,  1851.   $244875; 
what  was  due  l  Asa  S1.M.O03. 

OPERATION. 

Principal. 
3  9.8  1  3  Int.  from  June  4,  '48,  to  Sept.  9,  '49  . . .  1  yr.  8m.  5d. 

5  6  4.8  1  3  Amount  of  Principal  to  Sept.  9,  1849. 

1  1  4.2  0      1st  Payment. 

4  5  0.6  1  3  1st  Remainder,  forming  the  2d  Principal. 

1  8.4  7  5  Int.  from  Sept  9,  I  ay  1 5,  '50  . . .  8m.  6d. 

4  6  9.0  8  8  Amount  of  2d  Principal  to  May  15,  1850. 

•_\1  Payment. 

2d  Remaraderj  forming  the  3d  Principal. 

2  8.7  2  4  Int.  from  Ma;  - 1  . . .  lyr.  2m.  21d. 

4  1  9.5  2  7  Amount  of  3d  Principal  to  Aug  6,  lb 

2  4  4.3  7  5  3d  Payment. 

1  7  5.1  5  2  3d  Remainder,  forming  the  4th  Principal. 
1  5.8  5  1  Int.  from  Aug.  6,  '51,  to  Feb.  9,  '53  . . .  lyr.  6m.  3d. 

$  1  9  1.0  0  3  Amount  d  \  1853,  An*. 


L96 

I   l.     The  pupil  will  observ.  oration  is  performod  on  the 

slate  or  elsewhere,  only  the  results  being  here  written.     To  do  tho  work  hero 
would  take  up  too  much  space. 

Boston,  Mir.  26,  I860. 
value  received,  we  promise  to  pay  Stephen  C  Jones,  or 
ondred  forty-six  and  {\:\,  dollars,  on  demand,  with 

inter  Bruce  &  Davis. 

Indorsements:  July  20, 1860,  $54.75 ;  April  8,  1861,  $10; 
Sept  -Tan.   6,  1-  0.46  j  what  was  due 

on 

Principal. 
6.5  8  1  Int.  from  Mar  86,  '60,  to  July  20,  '60  . . .  3m.  I  Id. 

1  2.9  4  1  Amount  of  Principal  to  July  20. 
."»  1.7  Payment 

-.1  9  1    l-t  Remainder,  forming  tire  2<l  Principal. 

2  6.1  4  1    Int.  from  July  20,  '60,  to  Jan.  6,  '62  . . .  lyr.  5m.  16d. 

\  mount  of  2d  Principal  to  Jan.  6,  1862. 
1  6  5.9  6      Sum  of  2d,  3d,  and  4th  Payments. 

1  o  8.3  7  2  2d  Remainder,  forming  the  3d  Principal. 

3.0  6  2  Int.  from  Jan.  6,  '62,  to  May  2,  '62 . . .  3m.  26d. 

S  1  6  1.4  3  4  Amount  due  May  2,  1862.  An<. 

Ron  2.  !vcd  by  the  rule,  each  payment  being  greater  than  tho 

interest  which  had  arisen  ou  the  principal  at  the  time  of  the  payment ;  but 
in  Kx.  ')_»  it  is  found  by  trial  that  the  2d  and  3d  payments  were  less  than  tho 
interest  duo  on  the  principal  at  the  time  of  the  payments,  and  .-.,  in  accord- 
ance with  the  exception  in  tho  rule,  the  interest  is  cast  on  the  2d  principal, 
191,  from  July  10,  1890$  to  Jan.  6,  1^62,  tad  then  the  sum  of  the  2d, 
<l  4th  payments  is  taken  from  tho  amount  of  the  2d  principal. 

53.  $186.96.  12,  1860. 

value   received,  wo,  jointly  and  severally,  pronii 

Of  Order,    four   hundred    eighty-six    dollars   and 

md,  with  interest       Jam  tee, 

John    Di 
.    1861,   $154.87  i    Dee.    6,   1861, 
|100;  what  is 

v.ril  2  1.  18 


CAGE. 

1167.42.  Providence,  April  15,  1858. 

For  value  received,  I  promise  to  pay  A.  B.,  or  order,  one 
hundred  sixty-seven  and  ^0  dollars,  in  6  months  from  date,  with 
inter  C.   D. 

brDOBI  :    May   21,    1859,   $42.18;   July    17,    18G0, 

$6.25  ;  Sept.  9,  1860,  $48.16 ;  Jan.  27,  1861,  $27. 17  |  what  was 
doe  April  15,  L86S  Aas.  |72.072, 

55.  $172.76.  1.  1SC0. 

For  value  received  <»f  Walter  Willis,  1  promise  to  pay  bin 
Ins  order,  four  hundred  seventy-two  dollars  and  seventy- 
in  dl  DMQthfl.  from  date,  with  interest  at  7  per  cent,  afn -rwards. 

Samuel  Johnson. 

EmKMM  :  April   10,   1861,  $127- A  !  B  |   Nov.  28,1861, 

,7:M;  April  15,  1862,  S223.081 ;  what  was  iM   Nov.  13, 
1862? 

•00.  Andover,  Aug.  6,  1858. 

For  value  1  promise  to  pay  to  the  Trustees  of 

Phillips  Academy,  or  their  order,  in  Andover,  the  sum  of  fifteen 
hundred  doBai  y<ar  from  the  first  day  of  October.  A.  1  >. 

eighteen  handled  and  fifty-eight,  with  interest  to  he  paid  on  the 
first  day  of  April,  A.  D.  eighteen  hundred  and  fifty-nine,  and 
then  ird  half  yearly,  at  the  office  of  the  Treasurer  of  the 

said  Trustees  in  Andover.  ■   J.  S.  Pay  well. 

In  presence  of 

J.  L.  Tki  man. 

bn>OB8IMKKT8:  April  1,  1859,  $58.75 ;  October  1,  1859, 
$1  17>;  Nov.  1,  l>  |  Feb.  1.  I860,  $100;  April  1,1860, 

7.614;  what  wi  I  ily  1,  1860?  Ans.  $1065.75. 

239.  The  rule  given  in  Art.  238  is  the  one  adopted  by  the 
United  States  Courts  and  most  of  the  State  Courts;  but,  when 
settlement  is  made  within  a  year  after  interest  commences,  it  is 
customary  to  adopt  the  following 

838.  Where  is  the  work  performed?    Why  not  in  the  book?    239.  What  ml« 
is  usually  adopted  when  the  time  in  a  year  or 


IN  : 

r..      1.    Find  the  amount  of  the  principal  from  the  time 
<"st  commenced  to  the  tun 

from  the  time  of -payment 
to  the  time  of  settle//, 

with  their  interest  from 

U  of  the  principal 

»vo  rule  is  often   used    whftterer  maybe  the  timo;  hut 
for  Ion;;  periodl  it  is  manifestly  unjust,  for  by  it  the  debtor,  by  merely  paging 
■  annuaUg  at  6  per  cent,  will  In  Ian  than  M  1  hi*  entire 

nd   not   only  so,   the  person  who  loans  the  money  will  actually 
!  to  the  one  who  borrows. 

Bos{on,  Ma*  Iff,  1861. 
For  value  received,  I  promise  to  pay  to  Samuel  Adam?,  on 
demand,  three  hundred  eighty-seven  and  y0*y  dollar.-,  with  inter- 

IIf.nky    Phillips. 

nts:  July  21,  18G1,  $75;  Oct.  10,  1861,  $125; 
24,  1862,  $50 ;  what  was  due  at  the  time  of  settlement^ 
May    K»,  1862? 

SOLUTIOX. 

Principal,  $  3  8  7.7  5 

Latere**  of  Principal  for  1  year,  2  3.2  6  5 

Amount  of  Principal,  $  4  1  1.0  1  5 

Let  Payment,  7  5. 

Int.  of  1st  Payment  from  July  21,  9m.  24d.,  3.6  7  5 
2d  Payment,                                              12  5. 

Int.  of  2d  Payment  from  Oct.  10,  7m.  5d.,  4.4  7  9 

Payment,  5  0. 

Int.  of '3d  Payment  from  Feb.  24,  2m.  2  Id.,  0.6  7  5 

Bom  of  Payments,  with  their  Interest,  2  5  8.8  2  9 

Sum  due  May  15,  1862,  A  $  1  5  2.1  8  6 

58.  A  note  of  $2500,  dated  June  4,  1861,  has  the  following 

Im  Sept     1.    1861,  >;    Dec.   24,    1S61, 

S84f.  ■  .   18,  1862,  $362.63 1    what   was   doe    May   12, 

1862?  Ana.  $821.5*9. 

vi39.   Is  this  rah  jii-t  for  long  periods  of  time!    Why  not? 
17* 


198 

240.    Many  business  men,  In   computing  the    interest   on 
note.*,  adopt  the  following 

Kn.r.     Find  the  interest  of  the  principal  for  a  year  ;  also  of 
each  payment  made  during  the  year  from  the  time  of  payment  to 
the   end  of  the  year.      Then  subtract  the.   sum  of  the  paym 
together  with  their  interest,  from  the  amount  of  the  principal, 
the  remainder  is  a  new  pr inn 'pal.  with  which  proceed  for  another 
year,  and  so  on  to  the  time  of  settlement. 

59.  A  note  of  $1500,  dated  July  25,  ls.V.),  has  the  following 
iMDOftJ  :.  IS,  1859,  $100;  Jan.  25,  I860,  $300- 

19,  I860,  $260;    Dec  15,   I860,  $235]  Aug.  13,  1801, 

:   what  wa>  due  June  16,  If 

SOLUTI" 

jut  of  Principal  to  July  25,  'GO,  lvr.,  (    $  1  5  9  0. 

It  Paym  $100. 

Int.  of  U  Puv't  t«»  July  25,'CO,  10m.  12d..  5.2  0 

2d   Payment,  3  0  0. 

lot  Of  W  Payment  to  July  25,  'CO,  Cm.,  9. 

Sum  of  1st  and  2d  PayV.  with  Int.,  4  1  4.2  0 

Ipfl  Remainder  or  2d  Principal,  1  1  7  5.8  0 

Int.  of  2d  Principal  to  July  25/61,  lyr.,  7  0.5  4  8 

Amount  of  2d  Principal  to  July  25,  'CI,  1  2  4  C.3  4  8 

3d  Payment,  6  0. 

Int.of*  1  Pay'ttoJaJy25,'61,10nt,  12.7  5 

40)  P.-mmn't,  2  2-".. 

Int.  of  4th  Pay't  to  July  25,  '61,  7m.,  7.8  7  5 

Sum  of  3d  and  4th  Pay'K  with  Int.,  4  9  5.G  2  5 

Remainder  or  .°>d  Principal,  7  5  0.7  2  3 

[nt.  of  3d  Prin.  to  June  1 8,  '62,  1  Om.  I  3  9.7  8  8 

Amount  of  3d  Prin.  to  June  13,  18C2,  7  9  0.5  1  1 

5th  Payment,  $3  00. 

Int.  of  ;»th  Pay't  to  June  13,  'G2,  10m.,  1  5. 

5th  Payment,  with  its  Interest,  3  15. 

Sum  doe  at  settlement,  June  13,  'G2,  An-..  $  4  7  5.5  1  1 

"40.    Third  rule  for  computing  interest  on  notes? 


60.  A  note  oi  I  ' -  i  M.iv  15,  t859,  has  the  following 

June   1,  IS59,  $100;  July  7,  1860,  $100; 
1>       ;  '    I860,  |50  j  June  7,  18G1,  $100; 
•;  what  was  due  July  15,1863? 

Ans.  $302,011 

Note.  There  is,  perhaps,  no  other  operation  in  Practical  Arithmetic  in 
which  accountants  differ  so  much  as  in  the  mode  of  computing  interest.  All 
the  methods  are  based  upon  the  principles  developed  in  the  preceding  pages, 
and  it  is  b*  is  no   plan,  universally   applicable,  which   is  more 

and  simple  than  the  foregoing'.     The  solution  BftJ  usually,  however, 
be  much  shortened,  as  in  the  following  Article*. 

principal  advantage  arises  from  the  best  divisions  of  timo.  Facility 
in  making  the  best  divisions  can  be  easily  acquired  by  practice,  and  to  one 
having  frequent  occasion  to  compute  interest  the  attainment  is  of  great 
importance. 

241.     The  interest  of  $1  for  6  days,  at  6  per  cent.,  is  1  mill. 
Tlif  interest  of  $1    lor  Ml  times  6tl.  —  G0d.  =  2m.  is  1  cent. 
The  interest  of  $1  lor  ten  times  2m.  =  20m.  =  lyr.  8m.  is  1  dime. 
The  interest  of  SI  for  ten  times  20m.  =  ICyr.  8m.        ifl 
So  the  interest  i  f,  or  $1000,  for  the  same  times,  is  2,  3,  or 

1000  mills,  cents,  dimes,  or  dollars.     Thus  we  see  that  any  nuni- 
ber  of  dollars  expresses  its  own  interest  in  mills,  cents,  dim. 
dollars  for  the  above-mentioned  times,  and  hence,  to  know  the 
Interest  it  is  only  necessary  to  determine  the  place  of  the  decimal 
point. 

01.   What  is  the  interest  of  $324  for  93  days  ? 

OPERATION. 

I...  %     =  ht  for  3  Od.  ,AU,  h.ke   ex?ra.Ple3  can   be 

_  _  ,  solved   in  a  similar  manner. 

.10  2  =  Int.  lor     3d.  Henoe> 


=  Int.  for  9  3  d.,  Ans. 
242.     To  compute  interest  at  6  per  cent,  for  months 
and  <1 

IIui.k.     Move  the  decimal  point  in  the  j)rincij)al  two  places  to- 

ifl  of  different  modes  of  computing  interest?     What  of  the  best  divi- 
sion of  rin  .  '     Ml  1.    Any  sum  of  money  expresses  it*  own  interest  at  • 
cent,  fo*  what  times? 


200  PERCENTAGE. 

tcard  the  left,  and  the  result  will  be  the  interest  for  two  months 
or  sixty  pays.  Move  the  point  three  places  toward  the  lej),  and 
the  result  will  be  the  interest  for  ux  lays.  Then  tufa  such  muU 
iij)l<s  and  aliquot  parts  of  these  results  as  the  given  time  may 
require,  and  the  sum  of  these  trill  I><  the  interest. 

Proof.  Divide  the  computed  interest  by  the  interest  of  tht 
principal  for  one  month,  and  the  quotient  should  be  the  number  oj 
months  ea  in   the  example  ;  or,  divide  by  the  interest  for 

one  day,  and  die  quotient  should  be  the  number  of  days. 

ii:  1.  This  is  the  most  simple  modo  of  proof,  and  applies  to  all 
rules  for  computing  iutcrest.  The  Problems  in  Interest,  page  203,  furnish 
other  methods  of  proof. 

:  ■  2.  In  computing  interest  it  is  customary  to  consider  30  days  a 
month  tad  IS  nootfai  ■  year,  and  ,\  the  computed  interest  tor  12  times  30 
days,  or  360  days  (i.e.  for  g|j$  =  j§  of  a  year),  is  truly  the  interest  for  a 
whole  year.  Thus,  the  computed  interest  for  any  number  of  days  is  ^  too 
large  and  it  must  .'.  be  diminished  by  7*3  of  itself  to  find  the  true  rati 
As  UD  ially  computed  for  months  and  days  the  difference  U  slight, 

and,  in  course  <  :  is  seldom  considered  ;  but  in   England,  and  in 

dealing  with  the  United  States  Government,  it  is  customary  to  eompute  true 
interest. 

What  is  the  i:  I  for  7  months  an<l  3  »!. 

$7.2  0  =  Int.  for  2m. 

2  1.6  0  =  Int.  for  6m.        as  3  times  2ra. 
3.6  0  =  Int.  for  lm.        =  $  of  2m. 
.3  6  =  Int.  for  L  =  \  of  6 


6  as  Int.  for  7m.  3d.,  Ans. 
Troof.     The  interest  of  the  principal  for  1  month  is  S3. CO, 
and  the  Ans.  to  the  example  is  $25.5fi ;  .-.  the  time  in  months 
25.50  -r-  $3.00  =  7.1ra.  =  7m.  3d.,  the  time  given  in  the 
example. 

What  is  the  interest  of  $1260  for  75  days? 
$1  2.60  =  Int  for  0  0  d. 

3.15  =  Int.  for  1  5d.  =a  \  of  C  0  d. 
|  1  5.7  5  =  Int.  for  7  5  d.,  Ans. 

!•    Rule  for  computing  interest  for  months  and  days,  at  6  per  cent.?    Proof* 
Notel?    Vote  2! 


I.\ 

*J  I't.      TkrM  dOft  k    f  ,  <>f'a  month,  .*.   1V,  of  llio   im»TiM  of 

$1,  or  any  other  sum,  for    1    month.  i>  tin-   interest utf  the  *am« 

sum  for  9  day-.      Iii    lik--   mannrr,  ,V,  of  the  intnv-t  of  any  sum 

fur  fl  r  •/  months   is   the    interest  of  the  sa/ne  sura   for 

Ws  :is  many  days. 

64   What  is  the  intm-t  of  |7W  for  2m.  Cd.? 

OPERATION. 

$  7.6  5     =  Int.  for  2m.,  f.  e.  for  C  Od. 

5  =  Int.  for  ^  of  CM.,  i.e. 

$8.4  1  5  =  Int.  for  1.,  Alls. 

C5.  What  is  the  interest  of  $845  for  6  days? 

84.3  mills  ss  $.845,  Ans. 
G6.  Whsi  is  tbfl  interest  of  $345  for  2  months? 

845  cents  =  $8.45,  Ans. 
•  for  lyr.  8m.  ? 
Ten  times  845  cents  =  $84.50,  Ans. 
G8.  What  is  fee  interest  of  $845  for  lG§yr.? 

Ten  times  $84.50  =  $845,  Ans. 
Note.      The  pupil  will  observe  that  merely  changing  the  position  of 
•imal  point,  as  in   the  four  preceding  examples,  BJfoSi  the  interest  of 
any  sum  lor  G   days,   for  2  months,  for  1   year  and  8  months,  or  for  16J 
yeur>. 

G9.  What  is  the  interest  of  $845  for  lyr.  10m.  Gd.  ? 

OPERATION. 

$  8  4.5  0     =  Int.  for  lyr.        8m.,  i.  e.  for  2  0m. 

8.4  5     sa  Int.  for  ^  of  20m.,  i.  e.           2m. 
.8  4  5  =  Int.  for  £  of    2m.,  i.  e.         Gd. 

$  9  3.7  9  5  =  Int.  for  2  2m.  Gd„  Ans. 

What  is  fee  interest  of  $348  for  22  days? 

3  )  $  3.4JS     =  Int.  for_6_0  days. 

LI  6  r=  Int.  for  2  0  day s. 
.116  ss  Int.  for  2  days. 

$  lj  7  6=5  Iut.  for  "22  days,  Ans. 

«_ — _ —  1  ■ ■ 

'444.  One  tenth  of  thf  jnttrest  of  any  sum  for  any  number  of  months,  is  the 
Interest  of  the  same  sum  for  how  many  day*'  Bail  Ibff  •!•  ••> 1  mining  the  brfpfff 
of  any  sum  for  6  day*?    For  2  months!     For  lyr.  8in.'  Siu.f 


202  KNTAGE. 

71.   What  Ki  tlie  interest  of  S412  for  5m.?        Ans.  $10.30. 
7  2.  What  ii  ill*.  interest  of  1 12  for  2m.  22d.  ?  574 

What  ia  the  interest  of  $54  for  22d.  ?  Ans.  $.198. 

7  I     Wbal  ifl  the  interest  of  $2148  for  3m.  10d.? 
.   Wliat  ii  the  interest  I  C  lyr.  K'm.  i),].? 

the  interest  of  $173  for  1  yr.  8m.  ? 

944.  In  some  Statei  interest  ii  allowed  on  the  annual  In- 
terest of  the  principal  which  is  due  and  unpaid,  if  the  note  is 
written  "  with  interest  annually."  Sndi  exanipl.  I  may  be  solved 
by  CO  est  on  th>  wkoh  time  and  on 

each  year's  interest  for  the  titne  it  is  due  and  unpaid ;  but  the  fol- 
towSn  '■■  il  mode  of  computing  M  annual  interest  "  will 

be  of  service  to  the  business  man. 

R'  >m  the  principal  for  the  given  num- 

ber <  \rs  ;  on  this  interest  find  the  interest  for  half 

of  the  years  less  one,  and  the  months  and  days  ;  and  this  latter  in- 
terest is  the  J. x CESS  OF  t  for  the 
I  time.      To  this  excess  add  the  interest  on  the  principal  for  the 
whole  time,  and  the  sum  is  the  annual  interest  for  the  given  timu. 
77.   What  ii  the  annual  interest  of  $800  for  B 
$8  0  0,  Principal. 

J8  0 sb 8imple  Tnt.  of  SI  for  5  years. 

2  4  0.0  0  ss  Simple  Int.  of  $800  for  5  years.    *     j 
.12        =  Simple  Int.  of  $1  for  2yr.  i.  e.  for  —5-  =  2yr. 


2  8.8  0  =  Excess  of  annual  over  simple  Ink  of  $800  for  Syr. 
2  4  0        =  Simple  Int,  of  the  principal,  as  above. 

$  2  6  8.8  0  =  Annual  Int.  of  $800  for  5yr.,  Ans. 

78.  What  is  the  annual  interest  of  $600  for  6yr.  4m.  18d  ? 

Solution.  The  interest  of  $G00  for  6  years  is  $216;  the 
interest  of  $216  for  L  of  (6  —  1)  yr.,  increased  by  the  months  and 
days  viz.  8$yr.   4m.   18d.,  or  2yr.   10m.   184.  ia  $87,868,  and 

this  is  the  excess  of  the  annual  over  the  simple  interest  of  SC00 
for  6yr.  4m.  18d.  To  this  add  the  interest  of  $600  for  Gyr. 
4m.  18d.,  viz.  $229.80,  and  we  have  $267,168,  the  annual  int. 

344.    Rule  for  computing  annual  i    •  _    ... 


20S 

the  annual  interest  of  $402.84  for  7yr.  8m.  Gd.  ? 

An-.   $_'- 

80.  What  i>  the  excess  of  annual  over  simple  interest  of  $2o0 

...  *4<U  111.925. 

81.  What  is  the  amount  of  $325,  at  annual  interest  for 
flm.  i:»d.  ?  \  .13. 

What   is  the  amount  of  S4G92.80,  at  annual   interest   tor 

Problems  in  Interest. 

\M.>.  In  every  example  in  interest  there  are  four  elements 
or  particulars  which  claim  special  attention,  viz.  Principal,  Rate, 
Time,  and  Interest,  any  three  of  which  being  given,  the  other  can 
be  found. 

To  find  the  Interest  when  the  Principal,  Rate,  and  Time  are 

I,  has,  thus  tar,  been  the  object  of  our  discussion. 
Tin-  other  branches  of  the  subject  give  rise  to  the  following 
problems : 

216.  Problem  1.  Principal,  Interest,  and  Time 
given,  to  find  the  Rate. 

.  1.  At  what  rate  per  cent,  must  $300  be  put  on  interc-t  to 
-18  in  2  years? 

w.ysis.  $300,  at  1  per  cent,  will  gain  $6  in  2  years; 
.*.,  to  gain  $18,  the  rate  BUUt  be  the  quotient  of  $18  -j-  $6=3. 

Hence, 

i .v..  DMdi  the  given  interest  by  the  interest  of  the  princi- 
pal, for  the  given  time,  at  1  per  cent.,  and  the  quotient  will  be  the 
rate. 

At  what  rate  per  cent  must  $1 12  be  put  on  interest  to 
gain  $21.30  in  3  Jtt  Am, 

3.  If  S3G  gain  S7.5G  in  3  years,  what  is  the  rate  per  cent.? 

4.  If  $300  gain  $43.80  in  2yi\,  what  is  the  rate  per  cent.  ? 

345.  How  many  particulars  claim  attention  In  an  example  in  interest* 
What  are  they'  How  many  of  them  are  given?  910.  Object  of  Prob.  1? 
Rule? 


247.   Problem  2.    Principal,  Internet,  and  Bale  gii 

to  lad  the  T 

.1.  far  what  time  roust  $200  be  on  interest  at  B  pet1  cent 

an  $3G? 

Analysis.     $200  in  1  yoar,  at  C  per  cent.,  will  pain  STJ  ; 
in  $86,  the  time  in  years  must  be  the  quotient  of  $.3  <'»  ~  $12 
=  3.     Hence, 

Rule.    Divide  the  given  interest  by  the  interest  of  the  prhulpal 
for  one  year  at  the  given  rate,  and  the  quotient  will  be  the  time. 

I  low  long  must  $254  be  on  interest  at  5  per  cent,  to  gain 
$44.45?  ^yr.  =  «Yr.  6m. 

1  low  long  must  $75  be  on  interest  at 
SO?  Ans.  2.63  Jyr.  =  2yr,  7m.  18d. 

1.  How  long  must  $200  be  on  interest  at  I  Si  to 

nt  to  $286? 

faff  what  tinn-  must  $72  be  put  to  iutrn-t  at  ft]  pet  cent, 
nount  to  $87.30? 

I  'or  what  time  must  $1000  be  put  to  interest  at  9  per  cent. 
to  gain  $247.50  ? 

7.  How  long  must  $100  be  on  intend  at  5  pet  <rnt.  to  gain 
$100?  *  20  years.* 

Notb.    $100  in  1  year,  at  5  per  cent.,  will  pain  $5 ;  .*.,  to  gain  $100,  the 
time  in  jean  must  be  the  quotient  of  $100  -r  $5  =  20  ; 

To  find  the  tim>  |   any  sum  trill  double  itself  at  i 

rate  per  cr/tt..  diride   100  by  the  rate,  and  the  quotient  will  be  the 
in  years. 

8.  In  how  many  years  will  $50  amount  to  $100,  it  being  on 
Interest  at  8  per  cent?  Ans.  12yr.  Cm. 

long  will  it  take  any  sum  of  money  to  double  itself 
on  interest  si  •  t.? 

1<».  In  what  time  will  a  sum  of  money  triple  itself  on  interest 
at  5  per  cent? 

247.    Prob.  2?    Rule?    Rule  for  finding  the  time  in  which  auy  principal  will 
double  at  any  rate  per  cent.? 


^ils.     PftOBUttft.     Interest,  Time,  and  B 
find  the  Pkiwipal, 

Ex.  1.   What   principal,  at  6  per  cent.,  will  gain   $18  in  1  jr. 
Cm.? 

Awi.v-n.      si.   in    lyr.    Cm.,  at    G  per  cent.,  will   jrain 

.-.  the  principal  nui-t  hi-  the  quotient  of  $18  -~  .09  = 
Hence, 

Hl  i.k.     Divide  the  given  interest  by  the  interest  of  |1  for 
the  given  rate  and  tinn\  and  tk§  quotient  will  he  the  principal 

2.  What  principal,  at  6  per  cent.,  will  gain  $13  in  8 

Ant.  S83& 

3.  What  principal,  on  interest  at  8  per  cent,  per  annum,  will 

150  BemUannoaUj  ? 

4.  IS   endowed   a   professorship   with   a  salary    of  $2000  per 

annum;  what  sum  did  he  invest  at  G  per  cent.? 

i 

(a)  To  the  preceding  we  may  add 

Problem  4.    Amount,  Rate,  and  Time  given,  to  find 
the  Principal. 

Ex.  1.  What  principal,  at  5  per  cent.,  will  amount  to  $110  in 

Analysis.    $1  in  8  yews,  ^  ■">  p°i*  cent.,  amounts  to  $1.10; 

.-.  the  principal  innst  be  the  quotient  of  $110-4-  1.10  =  $100. 

ice, 

\U  :  'de  the  given  amount  by  the  amount  of  $1  for  (he 

and  time,  and  (he  quotient  will  be  the  principal. 

2.  What  principal,  at  6  per  cent.,  will  amount  to  $130.39  in 
8  mouths  ?  Ans.  $125.37."-. 

What    principal,  at  8  DOT  cent.,  for  3  years,  will  amount  to 
10? 
1     What  is  the  interest  of  that  sum  for  Syr.  Gm..  at  8  per  cent., 
which  will,  at  the  |  find  time  amount  to  $240? 

FMtv  4'    MM 


206  PERCENTAGE. 


COMPOUND   INTEREST. 

949.  Compound  Interest  is  interest  on  both  principal 
and  interest,  the  latter  not  being  paid  when  it  becomes  due. 

The  principal  may  be  inrn  a-<  <1  l>y  adding  the  interest  to  it 
annually,  m ■mi-annually.  quarter!;  cording  to  agreement, 

and  the  creditor  may  receive  compound  Interest  without  1> 
liable  to  the  cl  amrj  (Art.  231),  though  he  cannot  leya/Iy 

collect  it  if  the  debtor  refuses  to  | 

250.    To  calculate  Compound  Interest : 

Rti  r.  Make  the  amount  for  the  first  year  or  specified 
time,  the  PRINCIPAL  for  the  second  ;  the  amount  for  the  second 
the  principal  for  the  third ;  and  so  on.  From  the  last  amount 
subtract  the  fik^t  im:in«:ipal.  and  the  remainder  is  the  com- 
pound  interest. 

\.  I.  What  is  the  compound  interest  on  $100  for  3yr.  3m., 
at  6  per  cent,  per  annum  ? 

OrKRATIOX. 

$100.  Ill  Principal. 

$100  X-0  6=  6.  Interest  for  1st  year. 

10  6.  1st  Am't  or  2d  Prin. 
$106            x-0  6    =      _6.3  6            Interest  for  2d  year. 

1  1  %  S  1  Ain't  or  3d  Prin. 

$  1  1  2.3  6       x  -0  6    =  [16        Interest  for  3d  year. 

1 1  9.1 0 1  6        3d  Am't  or  4th  Prin. 
$  1  1  9.1  0  1  6  X-0  1  5  =  1.7  8  65  2  4  Interest  for  3  months. 

1  2  0.8  8  8  1  2  4  4th  or  last  Amount. 
10  0.  1st  Principal. 

$    2  0.8  8  8  1  2  4  Com.  Int.  for  3yr.  3m. 
Note  1.    Find  the  amount  for  the  years  as  though  there  were  no  months 
In  the  given  time,  and  this  amount  is  the  principal  for  the  remaining  mor.tfis. 

9.  Compound  Interest,  what  is  It?  How  often  may  the  interest  b»  com- 
pounded? May  the  creditor  receive  compound  interest  if  the  debto-  c-nooses  to 
pay?  Can  lie  collect  it  if  the  debtor  refuses  to  pay?  250.  Rule  for  computing 
compound  interest »    Rule  when  there  are  months  and  days  in  the  given  time  r 


209 

2.  What  la  the  compound  inter.  Bbl,  it  4 

at  per  annum.  -089. 

3.  What  is  the  compound  interest  on  $500  for  3  years,  at  7 
snt?  •  $112.5215. 

4.  What    [fl   the  amount  of  $5000  at  compound  interest,  for 

4yr.  10m.  I2d.?  Ans.  $6640.< 

What    is    the  amount  of  $9000  at  COMpOUnd   interest  for 

3 1  ?  Ana.  «& 

the  compound  interest  of  $10000  for  2yr.  Cm. 
nt.?  Ans.  $1606.788. 

7.  Whftl  b  the  compound  interest  of  $10000  for  2yr.  Cm. 
18d.,  at  1  per  cent.  ?  Ans. 

8.  What  is  the  compound  interest  of  $10000  for  2yr.  6m. 
18d.,  at  8  per  cent.?  Ans.  $2177.216. 

B  2.     Four  per  cent  of  any  number  is  |,  and  8  per  cent  is  J  of  G  per 

rent  of  the  same  number,  but  the  compound  interest  of  any  sum  of  money 

at  4  per  cent,  is  fan  than  |  of  the  compound  interest  of  the  same  sum  for  the 

at.,  and  the  interest  at  8  percent,  is  more  than  J 

of  the  interest  at  6  per  cent.,  as  may  be  teen  by  example!  8,  7,  and  8. 

The  compound  interest  at  4  per  cent,  is  less  than  half  the  compound 
interest  of  the  same  sum  at  8  per  cent.,  because  the  base  of  percentage,  (i.e. 
the  principal,)  after  the  1st  year,  is  leu  in  computing  interest  at  4  per  cent, 
than  in  computing  it  at  8  per  cent.  ;  thus,  in  computing  interest  at  4  and  8 
per  cent,  the  1st  year  the  base  is  the  same,  and  one  interest  is  just  half  of  the 
other;  but  the  2d  year  one  base  is  SI 04  and  the  other  SI 08  ;  .'.  the  interest 
at  4  per  cent,  is  less  than  half  of  that  at  8  per  cent. 

9.  What  is  the  amount  of  $250  for  2yr.  6ra.,  at  3  per  cent. 
h  6m.,  compounding  the  interest  semi-annually  ? 

Ans.  '$289,818. 

10.  "What  if  the  interest  of  $36  for  lyr.  9m..  at  2  per  cent, 
per  quarter,  compounding  the  interest  quarterly? 

Ans.  $5,352. 

11.  What  is  the  compound  interest  of  $864.75  for  8yr.  8m. 
at  6  per  cent.?  Ans.  $208.1» 

12.  What  is  the  compound  interest  of  $327.54  for  4vr.  4ra. 


.    It  compound   interest  at  4  for  cent,  half  as  much  as   at  8  per 

Wh;' 


208  PERri;\i 

2«H«    Compound   interest  may  be   calculated  more 
expeditiously  by  means  of  the  following 

TABLE, 

Stowing  the  Amount  of  81,  £1,  etc.,  interest  compounded  annually  at  4,  5,  6,  7, 
and  8  per  cent.,  from  1  to  20  years. 


Yr. 

4  per  Cent. 

5  per  Cent. 

6  per  Cent. 

Cent. 

8  per  Cent. 

Yr. 

1 

1 

1.040000 

1.050000 

1.060000 

1.O7O000 

1.0801 

2 

1.081600 

1.102500 

1.123600 

1.144900 

1.166400 

2 

3 

I.114M4 

1.157625 

1.191016 

1.991 

1.959719 

3 

4 

1.169859—    1.215506+ 

-177— 

1.340796+ 

1.360489— 

4 

5 

1.216653—    1.27( 

9896— 

1  409589— 

1.469328+ 

5 

6 

1.265319+ 

1.418519+ 

1.500730+ 

-74+ 

6 

7 

1.315932—     1 .407100+ 

1.6057M  + 

1.713824+ 

7 

8 

1.368569+     1.477 

548+ 

1.7191964- 

0990+ 

8 

9 

1 328+ 

1.689479  — 

1.838459+ 

1 .999005— 

9 

10 

1.480244+    1.628895— 

D848 

7151  + 

2.158925— 

10 

11 

1.539454+     1.714 

1 .898299— 

9191889 

2.331639— 

11 

12 

1.601033+ 

2012196+ 

.  1  92— 

2.518170+ 

12 

13 

1.665074— '  1.8854 

»28+ 

2.409845+ 

2.719624— 

13 

14 

1.781676+    1.971 

2.260904— 

-534+ 

2.937194  — 

14 

15 

1.800944—    2.078928+ 

2.396558+ 

2.759032— 

3.172169+ 

15 

16 

*1+    8.181 

'164— 

3.425943— 

16 

17 

1.947900+    2.292018+ 

2.692773— 

3.158815+ 

3.700018+ 

17 

18 

2.025817—    8404 

1399+ 

3.379932+ 

3.996019+ 

18 

19 

2.106849+    2.526950+ 

3.025600— 

3.616528— 

4.315701  + 

19 

20 

2.191123+    2.653298— 

3.207135+ 

3.869684+ 

4.660957+ 

20 

Note.     The  interest  is  81,  XI,  etc.,  less  than  the  amount  in  the  above  table. 

13.  Wh9i  is  the  compound  interest  on  $600  for  20yr.  ? 

$  2.2  0  7  1  3  5        =  Int.  of  $1  for  20yr.  taken  from  the  Table. 
60m0 

$132  4.2  81000  =  Int.  of  $000  for  20yr.,  Ans. 

14.  What   is  the  compound  interest  on  $30  for  5yr.  6m.? 

$  1.3  3  8  2  2  6     =  Amount  of  $1  for  5yr. 
.0  3     =  Int.  of  $1  for  6m. 

.04014678 

.338226  =  Int.  of  $1  for  5yr. 

$.3  7837278     =  Int.  of  $1  for  5yr.  6m. 
30 


$1  1.3  5118340  =  Int.  of  $30  for  5yr.  6m.,  Ana. 


DI8C01  NT. 

15.  What  is  the  amount  of  $50,  at  7  per  cent,  per  annum,  for 
impound  inter- 

7  5  9  0  3  2     =  Amount  of  Si  for  15vr. 
50 

;7.9  5  1  6  0  0  =  Amount  of  $50  for  l-'yr..  An--. 

1G.  What  is  the  amount  of  compound 

interest,  for  18  year 

17.  What  h   the  compound  Interest  of  $75  for  ,  at  8 

ent? 
What   is  the   inter  00  for  9yr.  fat,  :it   1  per  cent. 

for  each  G  months,  compounding  the  interest  semi-annually? 

An-  ;25. 

19.  What  is  the  amount  of  $100  at  compound  interest  for  40 
year-,  at  7  per  cent,  per  annum?  Ans.  $1497.445. 

20.  What  is  the  amount  of  $100  at  compound  interest  for  30 
.  at  G  per  cent  per  annum? 

DISCOUNT. 

252.  Discount  is  an  abatement  or  deduction  made  for  the. 
payment  of  a  debt  before  it  is  due. 

The  PRESENT  worth  of  a  debt,  payable  at  a  future  time  with- 
out in  <  vidently,  a  sum  which,  put  at  legal  interest,  will 
amount  to  the  debt  at  the  time  of  its  becoming  due. 

The  drbt,  men,  i-  an  amount,  the  present  worth  is  the  principal, 
and  the  discount  is  the  interest  of  this  principal.     Hence, 

253.  The  rule  for  finding  the  present  wortli  is  that 
given  in  Prob.  4,  Art.  248,  viz. : 

ide  the  given  sum  by  the  amount  of  $1  for  the  given  rate 
and  time. 

T7te  discount  is  found  by  subtracting  the  present  worth  from 
the  face  of  the  debt. 

.     What  is  Discount!    PraMri  Worth ?    The  debt  Is  the  same  as  what  in 
Art.  C;  ttl     *53.    Rule  for  finding  present  worth? 

Discount!    Explain   I 

18# 


210  PER< 

L  1.  What  is  the  present  worth  of  $37.44,  due  in  8  mont 
"What  the  discount  ? 

OPERATION. 

Amount  of  $1  for  8m.,  1.0  4)  3  7.4  4  (3  6,  Present  worth. 

81  2 

$  3  7.4  4,    Given  sum,  6  2  4 

8  6.0  0,   Present  worth.  6  2  4 

$  1.4  4,   Discount.  0 

t    What    k  the  present  worth  of  a  debt  of  $100,  payable  in 
J  «ar,  without  i  \\  'hat  the  discount? 

Ans.  Present  worth,  $94,389+;  discount,  $5.GG1 — . 

3.  What  is  the  present  worth  of  %Vi  I  _ M .  1 1  in.  ? 

Ans.  $1122.80. 

4.  What  is  the  present  worth  of  $14  1  .."»<>,  dm-  in    lyf 

Ana.  $131.32+. 
...   What   h  the  present  worth  of  $340.87,  dm    in  2jr. 

;— . 

What  is  the  discount  on  $456.25,  due  in  9m.  12d.  ? 
. 

7.  What  U  the  present  worth  of  $  I  tjt.  Cm.? 
WW  th.»  discount? 

8.  What  is  the  discount  on  $315,  due  in  1  year,  at  5  per 

haw   |   MJta  for  $1000,i  May  1,  18G3;  what 

int  shall  I  make  for  payment  to-  .  I86f,  money 

PH t  at   10  pV  cent.  pM  anmi 

is.     The  interest  on  the  present  worth  equals  the  discount  on  the 
debt. 

10.  What  is  the  interest  for  6  months  on  the  present  worth  of 
a  note  for  $350,  due  6  months  hence?  .  $10.19. 

11.  What  is  the  interest  for  a  year  on  the  present  worth  of  a 
note  for  $750,  due  1  year  hence? 

12.  I  have  a  note  for  $436,  payable  June  21,  1863;  what  is 
the  worth  of  the  note  to-day,  May  12,  1863,  money  being  worth 
8  per  cent,  per  annum  ? 

13.  What  is  the  discount  on  $896,  due  in  lvr.  8m.? 

14.  What  is  the  present  worth  of  $475,  due  in  2yr.  4m.  12d.  ? 


hank:  211 


BANKING   AND   BANK    DISCOUNT. 

253  a.    A  i  an  Institution,  ted  bylaw,  for 

the  kg  and  loaning  of  money,  deaDug  in  exchange, 

fbrnishing  a  currency  for  circulation,  et& 

charter  incorporating  a  bank,  defines  its  privileges  and 
limiti  Its  powers. 

The  Capital  Stock  of  a  bank  is  the  money,  paid  into  the 
bank  in  specie  by  the  stockholders,  as  a  basis  of  buss* 

i:  1.     Banks  arc  of  thrM  kinds,  viz.  :  Banks  of  Deposit,  Banks  of 
and  Hanks  of  Circulation. 
A  Bank  of  hi  i  and  takes  care  of  money,  subject  to  the  order 

of  the  depositor. 

A  Bank  of  DueotaU  Loani  money  upon  notes,  drafts,  and  other  securities. 
A    Bank  <>t"  Circulation    issues  its  own  bills  or  notes,  which  are  usually  re- 
deemable In  coin  at  the  hank  which  issues  them,  and,  because  redeemable  in 
■<tss  as  money  in  business  transui  n 
Hanks  in  this  country  usually  combine  the  threefold  office  of  deposit,  dis- 
count, and  circulation. 

i  I  2.     The  afTairs  of  a  bank  arc  controlled  by  a  Board  of  Directors, 
i  annually  by  the  stockholders  from  among  BBSSBSel 
The  President  and  Chatter,  appointed  by  the  Directors,  superintend  the 

I  of  a  bank  and  sign  all  bills  which  it  issues. 
A  Bank  Check  is  an  order  for  money,  drawn  on  the  bank. 
The  ./be*  of  a  note  is  the  sum  for  which  it  is  written. 
The  maturity  of  a  note  is  the  day  when  it  becomes  due. 
In  most  of  U  note  IS  not  legally  due  until  three  days 

after  the  time  which  the  note  specifies  for  its  payment.  These 
three  days  are  called  days  of  grace,  A  note  matures  upon  the 
last  day  of  grace. 

i:  3.     When  a  n  due  on  Sunday  or  a  legal  holiday,  it  is 

y  payable  on  the  preceding  day. 

25  J  a.    What  is  a  Bank?     What  of  its  privileges  and  powers?    What  is  the 
Capital  Stock  of  a  Bank?    Banks  are  of  how  many  ktodtl     What?    The  i 
What  of  hanks  in  this  country?     Directors,  how  chosen?     Dul 
if  CaaUarl     A  Bank  Check,  what?    The  face  of  a  note?    The  in: 
What  of  days  of  |1  SI  does  a  note  mature?     What  of  SaaAftjS  and 

holidays? 


212 

Note  4.  A  note  made  payable  in  a  certain  number  of  days  is  not  due 
nntil  that  number  of  days  expire;  thus,  a  thirty  dim  t,ote, 

Jan.  31,  becomes  due  Mar.  5  (or,  in  ' :     .  ij,  but  |  note  made 

payable  in  a  certain  number  of  months,  nominally  matures  on  the  same  day 
of  the  month  that  it  is  dai>  when 

it  matures;  or,  if  there  are  not  so  many  days  in  the  month,  it  i 
the  last  day  of  the  month  ;  thos,  a  one  month  note,  dated  on  the  28th 
ary,  nominally  matures  Mfir.  28,  and  legally  m  :    31  ;  but  a  one 

month  note,  dated  on  Jan.  28  (except  in  leap-year)  or  on  Jan.  29,  Jan.  30, 
or  Jan  31,  nominally  matures  Feb.  28,  and  legally  Mar.  3. 

£.121  1>.     Entered  OO  money  borrowed  at  a  bank  is  paid  ? 
the  money  it  borrowed.     The  interest  1  in  advance 

the  face  of  a  note,  and  retained  by  the  bank  as  con  on  for 

the  money  borrowed,  is  called  Bank  Discount.     The  money  n  - 
•1  by  the  borrower  is  called  the  Proceeds  or  Avails  of  the 
note,  and  is  equal  to  the  face  of  the  note,  )em  the  interest     The 
note  is  said  to  be  discounted. 

To  find  the  bank  discount  and  the  proceeds  of  a  note, 
payable  at  a  specified  future  time,  without  inter- 

Kri.i.  1  Find  the  interest  on  the  face  of  the  note,  at  the 
given  rate,  from  the  time  of  discounting  to  the  maturity,  and  the 
result  will  be  the  discount. 

2.  Subtract  the  discount  from  the  face  of  the  note,  and  die 
remainder  will  be  the  proceeds  or  avails, 

..  1.     What    i>  the  bank  discount  on  a  90  days  note  for 
$368  ?     What  are  the  proceeds  ? 

$  8.6  8     =  Interest  for  6  0  days. 
1.8  4     =  Interest  for  3  0  days. 
.18  4  =  Interest  for      3  days. 
$  5.7  0  4  =  Interest  for  9  3  days,  1st  Ans. 
$868  —  $  5.7  0  4  =  $  3  6  2.2  9  6,  proceeds,  2d  Ans. 

2.  I  have  a  6  months  ncte  for  S7G8,  dated  May  12 ;  what  will 
be  the  avails  if  I  get  it  diseoun:  8? 

253a.  A  note  payable  in  a  number  of  days,  when  due?  In  a  number  of 
months,  when  due?  253b.  Interest  paid  at  bank,  when?  Money  received,  called 
what?    Rule  for  finding  bank  discount?    For  finding  the  proceeds  of  a  note? 


PANKP  213 

$7.6  8     =  Int. owl  for  2  m. 

1.5  8  6  —  Int.-n-t  for  1  2  d- 
$9.2  1  G  =  Discount 

$  7  G  8  —  $  9.2  1  G  =  $  7  5  8.7  8  1 ,  proceeds,  Ans. 

months  and  grace  from  May  IS  expire  Nov.  15.     From 
Sept  .15  is   -in.  12cL,  the  time  for  which  the  note  is 

inted. 

What  will  be  the  bank  discount  and  what  the  proceeds  on  a 
4  months  note  !'• 

4.  On  a  90  days  note  for  $1812,  at  7  per  cent.? 
In  a  G  months  note  for  $489,  at  5  per  cent.? 
G.  A  4  months  note  for  $629,  dated  Feb.  27,  was  discounted 
Apr.  I2j  what  were  the  proc- 

7.  What  La  the  difference  between  hank  discount  and  true  dis- 
count (Art.  252)  on  an  8  months  note  for  S4G00? 

m,  1.     When  a  note  hearing  interest  is  diseountcd  before  its  matur- 
ity, the  amount  of  the  note  at  vialurity,  rather  than  its/ace,  is  the  base  for 
.ring. 

8.  AVI  nit  are  the  proceeds  of  a  note  for  $10000,  payable  in  6 
months  and  bearing  interest,  if  discounted  2  months  before  its 
maturity  ? 

The  amount  of  $10000  for  6m.  3d.  is  $10305,  and  the   in- 
;  of  $10305  for  2m.  is  $103.05,  which  taken  from  $10305, 
leaves  $10201.95,  Ans. 

9.  What  are  the  proceeds  of  a  note  for  $6844,  payable  in  4 
months  and  bearing  interest,  if  discounted  1  month  after  date  ? 

Note  2.  Business  men  often  deduct  more  than  the  legal  rate  of  interest 
for  present  payment  of  a  bill  having  a  term  of  credit. 

10.  What  shall  I  pay  on  a  6  months  bill  of  $75,  if  5  per  cent 
be  deducted  for  cash  ? 

1 1 .  What  on  a  bill  of  $250,  if  8  per  cent,  is  deducted  ? 

353  c.  To  find  the  sum  for  which  a  note  must  be 
written  that  the  proceeds  may  be  a  specified  sum. 

1.   For  what  sum  must  a  45  days  note  be  written,  that 
the  proceed-  may  he  1240? 

M3b.    What  it  Note  1?    Not*  2? 


2\  \ 

opKRATiosr.  The  proceeds  of  SI  for 

$  1.0  0  0      45    days    and    grace,    are 

Interest  of  $1  for  48  days,    .0  0  8       $0.9 !>•/,  and  .-.  ti 

Proceeds  of  $1.  .9  9  2      the  note  muct  b' 

dollars   as   $0,992   is  con- 
$  2  4  0  -*-  .9  9  2  =  $  2  4  1.9  3  5,      tained   times  in  $240, 

Ans.]       $241,935.     Hfl 

Rule.  fhe  required  proceeds  by  the  proceeds  0 

the  given  rate  and  time,  and  the  quotient  trill  be  the  numb* 
dollars  in  the  face  of  the  required  note, 

J.   For  what  sum  must  a  3  months  note  be  given,  that  the  pro- 
ceeds may  be  $800  ? 

3.  A   t'armrr  sold  produce  for  which  1  .lays 

note,  which  he  immediately  had  discounted  at  the  bank.     The 
proceeds  of  the  note  were  $593.70  ;  what  was  its  h 

1.   A  DM  n  -hant  wishes  to  borrow  $1200  i  ,  for  90  days  ; 

what  shall  be  the  face  of  the  note,  the  rate  of  interest  being  7 
per  cent.  ? 

2»5  I.     CmnUkVOl  hi  security  against  loss  from  the  damage 
or  destruction  of  property  by  fire,  shipwreck,  or  other  spe< 
casualty  ;  or  from  loss  of  life  or  health  by  disease  or  accident 

£»?•?.     The  Premium  is  the  sum  the  influaDOe,  and 

b  usually  computed  at  a  en-tain  per  cent  on  the  sum  insured. 

•s  according  to  the  nature,  locality,  etc.,  of 
property,  or   the   age,  place   of  resid«-n  rf  the   p« 

injured;  also  according   to  the   length   of  time   for  which   th<> 
ity  is  given. 

o  property  is  so  hazardous,  that  insurance  companies  dec-lino 
taking  the  risk  at  any  per  cent 

2.10.     The  Policy  is  the  writing  or  record  of  the  contract, 
i  by  the  insurer  to  the  injured.     The  policy  specifies  the 
nature  of  the  ri>k,  and  names  the  hour  when  it  begins  and  ends. 


253  c.  To  find  the  face  of  a  note  such  that  the  proceeds  shall  be  a  specified 
sum,  Rule?  254.  What  is  Insurance?  255.  Premium*  How  computed?  Does 
the  per  cent,  rary  ?    Why  ?    256.   What  is  the  Policy  ?    What  does  it  specify  | 


*2o7.    If  property  ii  fully  insured  th<'  owner  is  tempted  to 

destroy  the   property,  and    secure   its  value    from    the    in-uianec 

company.    To  prevent  inch  fraud,  oompeoiej  will  usually  insure 

the  j  ,'  :  only  about  <|  or  $  its  value,  requiring  the  owner 

t-t  ri-k   the  reniainder.     The   same  property  may  be   insured   .'it 

d  different  offices,  by  consent  of  the  companies  inanring 

it,  but  not  bo  th.it  the  whole  .sum  insured  at  the  different  offices 

shall  exceed  that   per  cent,  of  its  value  which  a  single  company 
nstomed  to  insure. 

2*18.    To  calculate  the  premium  on  a  givon  sum  : 

I  v  i  1.1:.     Multiply  the  sum  insured  by  the  rate  per  cent.,  written 
decimally. 

k.     The  insured  usually  pays  :i  -ivtn  <\\m,  say,  Si. 25,  for  the  policy, 
in  addition  to  the  premium  of  a  certain  per  cent,  on  the  aum  insured. 

Ex.  1.  What  is  the  cost  of  insuring  $2500  on  my  house  for 
at  2  per  cent.,  the  policy  being  $1.25  ? 

OPERATION. 

$2  5  0  0  X  -0  2  =  $5  0.0  0,   Premium. 
1.2  5,   Policy. 

$5  1.2  5,  Ans. 

2.  What  is  the  annual  premium  for  insuring  a  manufacturing 
establishment  in  the  sum  of  §75000,  at  3  per  cent.  ? 

Ans.  $$260. 

3.  In  a  certain  house,  the  furniture,  worth  $3400,  is  insured 
Tor  $  its  value  at  1$  per  cent.;  what  is  the  premium  ? 

4.  The    Merrimac    Mutual    Fire    Insurance    Company    have 
insured  $2000  on  my  house  for  a  period  of  5  years,  at  $  of  1 

:   what  ■  the  OMt,  the  policy  being  $1.25  ? 
1  buy  a  house  for  $8000,  and  get  it  insured  for  $  of   its 
value  at  3  of  1  per  cent. ;  the  house  being  boned,  what  is  my 
What  the  loss  of  the  insut. 

,  $2040;  loss  of  C 

257.  1*  property  usually  insured  for  its  full  value?  Why  not?  May  it  be 
insured  at  more  than  one  office?  On  what  conditions?  208  Rule  for  com- 
puting premium'     C<Mt  <>f  p<> if 


216  PERCENTAGE. 

0.  What  Li  the  premium,  at  \\  per  r  insuring  $70000 

on  a  steamboat  and  cargo  from  Boston  to  Havre  ? 

7.  A  cotton  factory  worth  $2f>000,  and  the  machinery  and 
stock  worth  $35000,  are  insured  for  ^  their  value  at  3  \ 

what  b  the  premium  ? 

8.  "What  i-  the  annual  premium  for  insuring  $6000  for 
on  the  life  of  a  man  25  years  of  age,  the  rate  1><  ing  .97  of  1 
cent,  annuall  Ans.  $58.20. 

What  will  bfl  the  annual  premium  for  insuring  $8500  for 
10  years  on  the  life  of  a  man  30  years  of  age,  the  premium 

STOCKS. 

259.  The  Capital  or  Stock  of  a  Bank,  Railroad,  Insur- 
ance, Mining,  or  Manufacturing  Company,  or  other  Corporation, 
is  the  money  or  other  property  employed  in  transacting  the  busi- 
ness of  the  Company.  City,  State,  and  Government  Bonds  are 
also  called  Stocks. 

260.  The  capital  or  stock  of  a  company,  is  usually  divided 
into  a  number  of  equal  parts,  called  shares,  and  the  owners  of 
the  snares  are  called  stockholders. 

20 1.  Shares  of  stock  are  bought  and  sold  like  any  other 
property.  The  nominal  or  par  value  of  a  share  of  stock  is  a 
fixed  sum  (in  most  companies  $100,  though  in  some  companies 
more,  and  in  some,  less),  but  the  market  value  varies,  according 
to  circumstances ;  as,  e.  g.,  if  a  company  is  prosperous,  and  its 
prospects  are  good,  its  stock  rises  in  price  ;  but  if  the  company 
has  been  unfortunate,  and  its  prospects  are  bad,  its  stock  declines. 

The  abundance  or  scarcity  of  money  also  affects  the  price  of 
stocks.  The  price  of  government  stocks  also  depends  upon  the 
state  of  the  country  as  to  peace  or  war,  the  prospects  of  the  sta- 
bility or  instability  of  the  govern r  .  etc. 

Note.  In  this  work,  SI 00  is  considered  the  par  value  of  a  share  of  stock, 
unless  some  other  sum  is  named. 

S39.  What  is  the  Capital  or  Stock  of  a  Company?  260.  Uow  divided? 
261.    What    i*   the   par  value  of  stock?    The  matkei  value,  how  doe*  it  vary! 


sto'  217 

£62.     If  a  share  of  stock  mO  nominal  value,  it  is 

raid  to  be  at  pari  tf  ll    Wlb    for  DOT*,  it   (a   at  a  premium,  in 
-  par  ;  if  it  sells  for  less,  it  is  crt  a  discount,  or 
;»ir. 
2623.    The    interest    paid    on    government   stocks,    and   the 
-  from  the  business  of  companies,  distributed  from  time  to 
among  the  stockholders,  are  called  Dividend** 
The  sums  of  money  occasionally  required  of  the  stockholders, 
to  meet  the  losses  or  expenses  of  the  company,  are  called  Assess- 
ments. 

26 1.  Assessments,  dividends,  discounts,  and  premiums  are 
percentages  on  the  par  value  of  the  stock  as  a  base.     Hence, 

Thoblem  1.  To  find  an  assessment,  a  dividend,  dis- 
count, or  premium : 

BULB.     Multiply  the  par  value  of  the  stock  by  the  rate  per 

.  written  decimally. 

Ex.  1.  The  directors  of  a  manufacturing  company,  wishing  to 
j"  their  works,  call  for  an  assessment  of  5  per  cent,  on  the 
capital  of  the  company  ;  what  will  be  the  assessment  on  $15000 
worth  of  the  stock  ? 

OPEi:  vi  ; 

$  1  5  0  0  0  ^ie  °Perat^on  *3  the  same  as  for 

q  5  computing  interest  for  1  year,  at  any 

— —  given  rate. 

S  7  5  o.o  o,  Aim. 

2.  The  Boston  and  Maine  Railroad  Company  paid  a  dividend 
of  4  per  cent.,  Jan.  1,  18G1  ;  what  was  paid  on  25  shares  of  its 
stock  ? 

OPEllATIOX. 

$100 

2  5  First  find  the  value  of   25  shares, 

g2  5  o  0  anc^  l*ien  comPute  the  dividend. 

S  1  0  0.0  0,  Ans. 

203.    When  is  stock  at  pur?    Abova  par?    Below  par?    201.    What  aro  dJv. 
lAcnds*    Arrestment*?    864.    Rule  for  computing  dividends,  assessments,  cto.f 
19 


218  PERCENTAGE. 

3.  What  is  the  discount  on  $1400  worth  of  stock  which  sells 
at  30  per  cent,  below  par?  Ans.  $420. 

4.  Suppose  the  New  England  Glass  Co.  Stock  sells  at  An 
advance  of  10  per  cent.,  what  is  the  premium  on  5  -Inns  at 
S-300  per  share  ? 

265.  Problem  2.  To  find  the  market  value  of  stock 
when  sold  at  a  premium,  or  at  a  discount. 

I'.x.  1.   What  is  the  market  value  of  $5000  worth  of  stock, 

at  a  discount  of  5  per  cent.  ? 

$5000 

9  5  Since  the  stock  sells  at  a  discount 

of  5  per  cent.,  $1  of  the  stock  sells 

\  a  a  a  *°r  :,,)  BeirtB»  ••  e-  tne  market  value 

45000  is  .95  of  the  par  value. 

$4  7  5  0.0  0,  At 

2.  What  is  the  market  value  of  f>  shares  of  Fitchburg  Rail- 
road Stock,  at  an  advance  of  2  per  cent.  ? 

OPERATION. 

$100 

6  First  find  the  par  value  of  6  shares, 

$  (To~0  nnc*  l*ien  •n('n,a5e  tne  F,ar  va-ue  Dv  tne 

I  q  2  2  per  cent,  premium,  i.  e.  multiply  the 

par  value  by  1.02. 
12  0  0  *  J 

600 


$  6  1  2.0  0,  Ans. 

similar  reasoning  holds  in  all  cases.     Hence  the 

Rule.  Multiply  the  par  value  of  the  stock  by  the  number 
which  represents  the  market  value  of  $1  of  the  stock. 

3.  What  shall  I  receive  for  12  shares  of  the  Andover  Bank 
Stock  at  l)  per  cent,  premium  ?  Ans.  $1308. 

1.  What  il  the  market  value  of  75  shares  of  Railroad  Stock 
at  a  discount  of  85  per  cent.  ? 

5.  What  is  the  premium  on  15  Shares  of  the  Western  Rail- 
road Stock,  at  18  per  cent,  advance? 

'463.    Rule  for  finding  the  market  value  of  stocks* 


219 

260.  Problem  3.  To  find  how  many  shares  of  stock 
may  be  bought  for  a  piven  sum. 

I.\.  1.  How  many  >hares  of  Railroad  Stock  may  be  bought 
for  $870,  when  the  market  price  i-  19  pi  r  cent  betow  par? 

OPERATION.  $1     of    Stork      El     WOlMh 

$8  7  0 -$-.8  7  =  $1000.  only  87  cents.-,  tfc 

$1000-f-$100  =  10,  Ans.  :-  .87,  viz. 

si11'''*,    i-     the     nominal 
•k  botgjkft.      Again  $1000  divided  by  3100,  the 
nominal  value  of   1  >har<\  gives  10  tharee,   Aus. 

2.  I  lew  many  -hares  of  the  Western  Railroad  stock  maybe 
purchased  for  $575,  when  it  is  worth  16  per  cent,  premium? 

<>ii  i.viiox.  $\     of     stock     is     worth 

$  5  7  5  -h  1.  1  5  =  $  5  0  0.  .  i  ">,   .-.    $578  ■*  1.15  = 

$5  0  0  -4-  $  1  0  0  ~  5,   An>.  $500,  is   the   nominal  value 

of   the    purchase.      A 
$500  -4-  $100  =  5,  the  number  of  shares  purchased     Hence, 

Kile.  1.  Divide  the  sum  expended  by  the  number  represent- 
ing the  marb  (  value  of  SI  of  the  stock,  and  the  quotient  is  the 
nominal  value  of  the  stock  bought. 

2.  Divide  the  nominal  value  of  the  purchase  by  the  nominal 
value  of  1  share,  and  the  quotient  is  the  number  of  shares  bought. 

3.  Now  many  shares  of  the  Exchange  Bank  Stock,  at  2">  pet 
cent,  premium,  can  be  bought  for  $1000?  Ans.  8. 

•1.  I  low  many  shares  of  Mining  Stock,  at  12  per  cent-  dis- 
count, may  be  bought  for  $2200  ? 

COMMISSION  AND  BROKERAGE. 

267.    Commission   or   Brokerage   is  the  compensation 

•  d  by  an  agent  for  transacting  certain  kinds  of  business, 

men,  e.  g.  at  collecting  and  loaning  money,  or  buying  and  selling 

<ks  etc. 

The  agent   is  variously  styled  as  factor,  broker,  collector,  cor- 

terekant,  i 

tor  finding  how  many  share*  of  stock  may  be  bought  for  a  given 
-407.    What   is  Comraiwsiou   or   Brokerage?     What   if  the  agent  styledl 


220  PKBCBBTi 

268*  Commission  or  Brokerage  is  a  certain  percentage  on 
Jhe  money  collected  or  expended.     Hence, 

Problem  1.  To  compute  Commission  or  Brokerage 
on  a  given  sum  : 

Rule.  Multiply  the  given  sum  by  the  rate  per  cent^  written 
decimally,  and  the  product  will  be  the  commission. 

Ex.  1.  What  shrill  I  pay  my  agent  for  selling  $4786  worth 
of  goods,  his  commission  being  4  per  cent.  ? 

$47  8  6  X  .0  4  =  $1  9  1.4  4,  Ans. 

2.  A  commission  merchant  sells  farm  produce  to  the  amount 
of  $1892;  what  is  his  rommis>ion  at  2  per  cent.? 

3.  The  taxes  in  the  town  of  B  for  1862,  are  $15000  ;  what 
is  the  cost  of  collecting  them  at  \  of  1  per  cent.  ?     Ans.  $7 

4.  My  agent  has  lent  for  me  $2124.  His  commission  is  \  of 
1  per  cent.;   what  shall  I  pay  him? 

5.  My  correspondent  in  Paris  has  bought  for  me  C  ball 
French  calico,  each  bale  containing  50  pieces  of  30  yds.  each, 
at  25c  per  yd. ;  what  Lfl  his  commission  at  $  per  cent.  ? 

6.  My  agent  in  New  Orleans  has  sold  for  me  400  pair-  of 
boots  at  $1.50,  400  pairs  of  shoes  at  75c,  and  500  pairs  do.  at 
$1  ;  what  is  his  commission  at  3  per  cent.,  and  what  shall  he 
remit  to  me  ?  2d  Ans.  $1358. 

269.  Problem  2.  To  find  the  commission  or  bro- 
kerage, when  the  agent  is  to  take  his  pay  from  the  sum 
remitted  and  invest  the  balance 

Ex.  1.  Sent  my  agent  in  London  $5100,  out  of  which  he  is  to 
take  a  commission,  and  invest  the  balance  in  goods.  What  sum 
will  he  invest,  his  commission  being  two  per  cent,  on  the  pur- 
chase, and  what  is  his  commission  ? 

$  5  1  0  0  -4-1.0  2  =  $  5  0  0  0,  Investment. 
100  —  $5000  =  $100,  Commission. 

Since  the  commission  is  2  per  cent,  on  the  sum  expended,  the 
agent  must  have  $1.02  for  every  dollar  he  pays  for  goods;  .*.  he 

*» » 

36S.    Rule  for  computing  commission? 


TA.V 

can  invest  as  many  dollar  mtnined  times  in  S")100, 

vi/.  95000,  Mid   thil  lubtracted  from  the-   SJ100  gives  §100  for 
■mmi»ion.      Hi 

BULB.     1.  D  m  sum  by  1  increased  by  the  deci- 

tsing  the  rate  per  cent. of  commission,  and  the  fju 
will  be  tit,-  sum  tn  lie  jfMN slrd. 

J.  The  sit//t  in  Listed  subtracted  from  the  given  sum  will  bare 
the  commission. 

2.  I  intrust  $10000  to  my  factor  in  New  Orleans  ft>r  the  juir- 
of  cotton.    What  sum  shall  he  inve>t  after  deducting  £  per 

cent,  commission  for  the  purchase,  and  what  are  hi-  i< 

Ant.   S-,'.,-"><».i.>o — ,  Invf-tiii.-nt ;    $49.75-j-.  Commission. 

8.  Sent  $10100  to  a  Boston  broker  for  the  purchase  of  bank 
stock.  The  brokerage  inr\  per  cent,  on  the  purchase ;  what  does 
bfl  pay  for  stock,  and  what  is  the  brokerage  ? 

4.  Sold  a  quantity  of  merchandise  for  my  employer  for  $5000. 
Alao  purchased  goods  lor  him  to  a  certain  amount,  and,  having 
calculated  my  commission  at  5  per  cent,  on  the  sale  and  3  per 
cent,  on  the  purchase,  our  accounts  balanced ;  what  did  I  pay  for 
the  goods  bought  ?  What  was  my  commission  on  the  sale  ?  On 
the  purcha  •» 

TAXES. 

270.  A  Tax  is  a  sum  of  money  assessed  upon  the  person, 
the  property,  or  the  income  of  individuals  by  the  authorities  of 
a  town,  county,  state,  or  other  section  of  a  country,  or  by  the 
national  government,  to  defray  the  expenses  of  government,  to 
construct  public  works  of  common  utility,  etc. 

271.  A  tax  on  property  \~  at  a  certain  per  cent,  on 

timafed  value  of  the  property. 

The  tax  on  the  person,  called  the  capitation  or  poll  tax,  is 
assessed  equally  upon  all  individuals  liable  to  pay  a  poll  tax.     A 

I  ii  called  a  pull. 

20tK     Ku!e    when   tin.-   commission    is  to    be    taken    from   the   nun    remitted! 
270.     What  is  a  tax?    By  whom  assessed?     For  what'    871.     How  is  the  tax 
on  property  assessed?    The  tax  upon  the  person,  called  what?     What  is  a  poll? 
19* 


222  PERCENTAGE. 

272.  Property  is  of  two  kinds,  viz.  real  and  personal  estate. 
Real  Estate  consists  in  immovable  property ;   e.  g.  lands, 

houses,  ?nills,  etc 

Personal  Estate  consists  in  movable  property,  as  money> 
notes,  cattle,  tools,  ba)ik  stocks,  railroad  stocks,  ships,  etc. 

273.  An  Inventory  is  a  list  of  articles  of  property,  With 
their  estimated  value. 

274.  The  method  of  assessing  taxes  is  not  the  same  in  all 
its  details  in  the  different  States,  but  the  essential  principles  are. 

In  some  of  the  States  the  tax  bill  is  so  made  as  to  show  the 
amount  of  tax  upon  the  real  (Mate  and  personal  property  sepa- 
rately ;  in  other  States  no  such  distinction  is  made 

In  Vermont,  each  taxable  poll  is  reckoned  as  so  much  property, 
say  $200,  and  DO  separate  poll  tax  is  calculated.  This  shortens 
the  operation  of  making  out  a  tax  lit. 

In  Connecticut,  personal  property  is  taxed  ju-t  twice  as  high 
as  real  estate  ;  thus,  if  A  pays  $80  on  a  farm  worth  $1000,  then 
B  would  pay  $f>0  on  $4000  at  interest. 

27»T.    In  Massachusetts,  the  assessors  are  required  to  as 
upon  the  polls  about  one  sixth  part  of  the  tax  to  be  raised,  pro- 
vided the  poll  tax  of  one  individual  for  town,  county,  and 
purposes,  except  highway  taxes,  -hull  not  exceed  $2.00  for  one 
year.     The  remainder  of  the  sum  to  be  raised  is  apportioned 
upon  the  taxable  property  of  the  town,  county,  or  state.     Hence, 

To  Assess  Tax 

Ki'LE.  Ascertain  the  number  of  polls  liable  to  taxation,  and 
take  an  inventory  of  the  taxable  property.  Multiply  the  sum 
assessed  upon  one  poll  by  the  number  of  taxable  polls, and  tubftxu  t 
the  product  from  the  swm  to  be  raised.  Divide  the  remainder  by 
the  taxable  property,  and  the  quotient  will  be  the  tax  upon 
MitUipfg  the  taxable  property  of  an  individual  by  the  number 
expressing  the  tax  upon  $1,  to  the  product  add  his  poll  tax,  and 
the  su?n  will  be  his  total  tax. 

273.  How  many  kinds  of  property?  What  is  Real  Estate?  Personal  Estate? 
273.  What  is  an  Inventory?  274.  Are  the  details  of  taxation  the  same  in  all 
the  States?    What  peculiarity  in  Vermont?    In  Connecticut?    275.    The  rule  in 


own  of  A  is  to  be  taxed  The  real  I 

of  the  town  la  valued  al  §500000  and  ike  personal  at  $500000. 
are  666  taxable  polls  each  of  which  is  assessed  $1.50. 

"What  is  the  tax  of  B,  whose  real  I  I  alm-d  at  $4000  and 

y  at  $8000,  and  who  pays  1  poll  tax  ? 
$1..'.<>  x  666  =  $999,  sum  assessed  on  th<-  polls. 

199  — 999  =  $5000,  sum  to  be  assessed  on  the  property. 
$500000  +  300000  =  $800000,  amount  of  taxable  property. 
$50  >000  =  6}  mills,  tax  on 

>00  -f  $8000  =  $12000,  B's  taxable  property. 
$12000  x  .006J  =  $75,  tax  on  B's  property. 
>  =  $76.50,  B's  entire  tax,  An*. 

Note.  To  save  labor,  (by  using  smaller  numbers,)  assessors  frequently 
take  6  per  cent  of  the  inventory  instead  of  the  entire  valuation  ;  but  the 
labor  may  be  lessened  still  more  by  taking  10  per  cent.,  as  in  Ex.  2. 

2.  The  town  of  F,  whose  valuation  is  $356400,  has  6  taxable 
inhabitants,  A,  B,  C,  D,  E,  and  F,  who  wish  to  raise  a  tax  of 
$1800.  The  taxes  of  the  several  inhabitants  are  for  the  number 
of  ]x>lls  and  the  property,  as  in  the  following 


INVENTORY. 


Names. 

Number 
of  Polls. 

Real  Estate. 

Personal 

ate. 

Total. 

10  per  Cent 

A 
B 

C 

1) 

i: 

F 
To: 

3 
2 

1 
3 
3 

24875 

1*469 

2842  1 
15860 
19933 

70405 
88460 
47628 
06486 

34867 

95280 
38460 
67090 
8491M 
15860 
5  1800 

952$ 

3846 
6709 

8491 
1586 
5480 

12 

108554 

247846 

356400 

35640 

The  tax  upon  eaeh  poll  being  $1.50,  what  per  ecnt.  i>  levied 

on  the  property, and  what  is  the  tax  of  A,  F>,  ('.  1).  E,  and  F  ? 

assessors  to  nve  labor*     What 
.  t  is  suggested?     What  is  the  object  Of  the  Tabk  I    Lxpiaiu  Ex.  2. 


224 


PERCENTAGE. 


la  calculating  a  tax  list  it  is  must  convenient  to  form  a  table 
showing  the  tax  upon  SI,  $2,  ■.  in  the  percentage  column, 

and  then  calculate  the  taxes  of  the  BevenU  inhabitants  from  the 
table  ;  thus,  in  solving  Ex.  2,  first  find  the  tax  raised  on  all 
the  polls  ($1.50  X  12  =  $18),  and.  having  deducted  this  from 
the  total  tax,  ($1800  —  $18  =  $1782),  divide  the  remainder 
by  the  assumed  percentage  of  the  taxable  property  in  town 
($1782  -i-35640  =  $.05),  to  find  the  tax  on  $1  in  the  percent* 


age  column.     Then  lorm  the 


TABLE. 


Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

Prop. 

Tax. 

8 

8 

8 

8 

8 

s 

8 

8 

1 

0.05 

10 

0.50 

100 

5.00 

iooo 

50.00 

2 

0.10 

20 

1.00 

200 

10.00 

2000 

100.00 

3 

0.15 

30 

1.50 

300 

15.00 

3000 

150.00 

4 

0.20 

40 

2.00 

400 

20.00 

4000 

►.00 

5 

0.25 

50 

2.50 

500 

25.00 

5000 

250. 

G 

0.30 

GO 

3.00 

GOO 

30.00 

6000 

300.00 

7 

0.35 

70 

3.50 

700 

35.00 

7000 

350, 

8 

0.40 

80 

4.00 

800 

40.00 

8000 

400.00 

9 

0.45 

90 

4.50 

900 

45.00 

9000 

450.00 

Now  to  find  A's  tax  from  this  table : 

OPERATION. 

Tax  on$9000  =  $45  0. 

"      "         5  00=         2  5. 
"      "  20=  1. 

«      "  8  = 

■      «      3  polls  =  •!..*>  <> 


A's  total  tax  =  $  4  8  0.9  0 


In  the  same  manner  the 
tax  of  B,  C,  etc.,  may  be 
found.  By  the  abov 
soiling  tin*  tax  is  found  to 
be  5  per  cent,  on  the  per- 
centage column,  or  i  per 
cent,  on  the  entire  taxable 
property. 


CUSTOM-HOUSE  BUSINESS. 

276.  Customs  or  Duties  are  taxes  levied  by  the  General 
Government  on  imported  or  exported  goods,  to  support  the  gov- 
ernment and  to  protect  home  industry. 


276.    What  are  Customs  or  Duties? 


225 
277.    All  g  ht  into  the  (Jailed  States  from  fbrtlgn 

countries  mii-t  be  landed  at  Berttdli  plaees   called  />o/7s  of  tutry. 

•  ach  port  of  entry  a  custom-house  ii  tttaMkhed  by  govern- 
ment, with  offloen  t<»  ( ■iiin])uii'  and  coiled  the  «luti«  -. 

All  duties  are   regulated  by  government,  and  are  different  at 
:t  times  aud  in  different  conn' 

Po  lirin^  in  iiiLTiluuKlise  secretly  and  without  paying  dm 
eaMvilsmw/ylinij,  and  j><  tie  liuLle  to  punishment  if  detected. 

£7S.     Tn\'N.vi;K  is  ■  tax  upon  the  vessel,  without  reference 
Cargo,  for  the    privilege   of  coining  into  I  port  of  entry. 
The  amount  of  tonnage'  depends  upon  the  size  of  the  i  ■ 

The  income  from   duties    and    tonnage   ifl   the   n-nnue  of  the 
rnment     Occasionally,  when   the   revenue   from  duties  and 
-   insufficient  to  defray  the   expenses  of  government, 
'   taxes  are   levied,  by  authority  of  our  national   con. 
upon  the  pertOO,  the  property,  and  the  incomes  of  the  inhabitants. 
£7!>.     Dutie-  are  either  ud  valorem  or  sj»cijic. 
An  An   vai.okkm  I)i  rv  is  a  certain  percentage  computed  on 
the  market  value  of  the  goods  in  the  country  from  which   they 
are   imported. 

A  Sj  i .<  ific  Duty  i>  a  certain  sum  per  ton,  gallon,  yard,  etc-, 
without  regard  to  the  cost  of  the  article. 

*£>0.    An  Invoice  is  a  list  of  the  articles  sent  to  a  purchaser 
or  agent,  with  the  pvices  annexed. 

Ad  valo,:i.m   Dities. 
2S1.    Problem  1.    To  compute  ad  valorem  duties: 
BULK,     Multiply  the  mst  of  the  goods  by  the  given  per  cent. 

Ex.  1.  What  is  the  duty,  ■(  •!«>  per  cent.,  on  25  cases  of  French 
broadcloths,  invoiced  at  $30000? 

S30000x.-i0  =  $1200  0.0  0,  Ans. 

Imported  goo<I.»,  where  landed?  A  customhouse,  what?  Smuggling, 
what?  878.  Tonnage?  Government  revenue,  how  obtained?  Direct  taxe«, 
when  levied?  *79.  How  many  kinds  of  duties?  What?  Ad  valorem  Dutie*, 
what?    Specific?    280.  An  In  voice.'    281.  Kule  for  computing  ad  valorem  duUeat 


226  PERCENTAGE. 

2.  What  is  the  duty,  at  25  per  cent.,  on  47961b.  of  Russia 
iron,  worth  10c.  per  lb.?  Ans.  Si  19.90. 

3.  What  is  the  duty,  at  3G  per  cent.,  on  an  invoice  of*  silks, 
which  cost  S57G5  in  Italy? 

4.  At  33£  per  cent.,  what  is  the  duty  on  an  invoice  of  Irish 
linen,  amounting  to  $13248? 

Specific  Duties. 

28£.  Specific  duties  are  computed  only  on  the  actual  weight 
or  measure  of  merchandise ;  hence  certain  allowances  are  made 
before  calculating  the  dot 

Leakage  is  an  allowance  of  a  certain  per  cent,  on  liquors  in 
Casks,  paying  duty  by  the  gallon. 

Breakage  is  an  allowance  of  a  certain  per  cent,  on  liquors 
in  bottl e& 

Draft  or  Tret  is  an  allowance  made  in  the  weight  of  goods, 
•  of  waste  or  refuse  matter. 

Tare  is  an  allowance  on  account  of  the  weight  of  the  box, 
cask,  bag,  etc.,  which  contains  the  goods. 

Gross  Weight  is  the  weight  of  the  article  before  any  of  these 
allowances  arc  made. 

Ni:t  WEIGHT  is  the  weight  of  the  merchandise  after  all  the 
allowances  are  made.     Duties  are  computed  on  net  weight. 

Note.  The  rates  of  draft,  tare,  leakage,  etc.,  are  regulated  by  law,  and 
ore  different  on  different  articles  and  at  different  times. 

283.    Problem  2.     To  compute  specific  duties. 

Ex.  1.  What  is  the  duty  on  10  casks  of  molasses,  containing 
65  gallons  each,  at  5  cents  per  gallon,  allowing  2  per  cent,  for 
leakage  ? 

OPERATION. 

G  5  X  1  0  =  C  5  0,  No.  gal.  in  10  casks. 
G  5  0  X  -0  2  =     1  3,  Allowance  for  leakage. 

G  3  7,  No.  gal  net. 

G  3  7  X  -0  o  =  3  1.8  5  ;  .-.  duty  =  $3  1.8  5,  Ans.     Hence, 

283.  Specific  duties,  computed  on  what?  What  is  Leakage?  Breakage* 
Draft  or  Tret  ?    Tare  PC  >  Weight  r 


MANGE. 

Deduct  the  legal  draft,  l**V*i  etc.,  from  the 

uiity  of  set  >tltij)h/  the  remaind 

the  </                     h  gallon,  pound,  yard,  etc..  and  the  product  will 
be  the  duty. 

2.  AVlint  is  the  doty,  at  4c  per  lb.,  on  500  bags  of  coffee, 
weighing  2001b.  each,  tan  I  per  ••'•in.? 

What  i>  tin*  doty,  at  6c  per  lb.,  on  300  boxes  of  figs, 
weighing  1121b.  etch,  allowing  lib.  draft  and  151b.  tare  on 
each  !  728. 

1.  What  is  the  duty,  at  15c  per  lb.,  on  -18  cheats  of  tea,  each 

weighing  66 IK,  draft  being  1  lb.  per  bos  and  tare  4  per  cent,  on 

mainder? 

5.  What   is   the   duty,  at  5c.  per  lb.,  on   800  bags  of  coffee, 

weighing  561b.  each,  draft  being  lib.  for  each  1121b.  and  tare 

t.  on  the  remainder? 

EXCHANGE 

08-1.  Ex<  u  \  n  . ;  i : ,  in  commerce,  is  a  mode  of  paying  debts 
due  in  distant  places  by  meAns  of  drafts  or  bills  of  exchange, 
without  the  cost  or  risk  of  transporting  specie. 

28o.  A  Draft  or  Bill  of  K\<  iiam.i;  if  a  written  order 
or  request  to  one  person  to  pay  to  another  a  certain  sum  of 
money,  and  charge  the  same  to  the  account  of  the  person  who 
makes  the  request. 

USfi.     The   Ma  ma  or  Drawer,  of  a  draft  or  bill  of  ex- 
change  \-  tlw  person  who  reqoests  another  to  pay;  the  Drawkf, 
person  who  is  requested  to  pay  ;  and  the  Path  is  the 
d  to  whom  the  drawee  is  requested  to  pay  the  money. 

*2**7.    To  explain  the  operation  of  exchange  and  show  it* 

benefits,  l<t  us  suppose  an  example:  A  of  Boston  owes  B  of 

ion  $1000,  and   C  of  London   owes    1)  of   Boston   $1000. 

A  and  C  can  each  pay  his  debt  by  sending  $1000  in  gold 

or  silver  and  paying  the  cost  of  shipment  and  insurance  ;  but 

283.  Rule  for  computing  specific  duties?  C84.  What  is  Exchange?  285.  A. 
Draft  or  Bill  of  Exohangef    286.    The  Maker  or  Drawer?    Drawee?    Pajaa? 


228  PSB   ENTACl. 

exchange  furnishes  ■  better  way.  Thin,  D  of  Bottoo  writes  a 
request  (bill  of  exchange)  to  C  of  London  that  he  would  pay  A 
of  Boston,  or  his  order,  $1000.  A  baji  this  hill  of  exchange 
of  D  and  pays  him  for  it  in  Boston  money,  endorses  the  bill  and 
sends  it  to  B  of  London,  who  presents  it  to  C,  and  (  pays  13  the 
$1000  in  London  money;  thus  A  and  C  have  paid  their  debtl 
and  B  and  D  have  1  their  dues  without  the  trouble 

or  risk  of  sending  a  dollar  in  money  or  merehandix-  aero-s  the 
Atlantic;  and  betides,  there  is  the  same  amount  of  money  in  both 
London  and  Boston  as  there  would  be  if  A  and  C  had  paid  their 
respective  debts  by  remitting  gold. 

288.  Some  bills  of  exchange  are  made   payable  at  ri 

i.  e.  as  soon  a3  they  are  presented  to  the  drawee ;  others  are 
made  payable  on  a  L'iv.-n   day  or  in  a  speeitied  time,  say  $ 
or  90  days  after  light     Usually  3  days  of  grace  (Art.  253a)  are 
added  to  the   time  specified  in  the  bill,  but  this  custom  is  not 
uniform  in  all  pla. 

289.  The  payee,  instead  of  reeeiving  the  money  from  the 
drawee,  may  tell  the  bill  to  another,  and  he  in  turn  may  sell  it 
again,  and  so  on  indefinitely.  Any  person  who  buys  the  bill  is 
called  the  Buyer  or  BuUfUli 

The  person  who  owns  the  bill  at  any  given  time  is  the 
Holder  or  Possessor. 

The  payee  and  the  several  buyers,  by  writing  their  names 
across  tin-  back  of  the  bill,  become  Indorsers,  and  responsible 
to  the  holder  for  the  payment  of  the  bill  at  maturity,  i.  e.  at  the 
time  when  the  bill  becomes  due. 

290.  Bills  payable  in  a  given  time  after  right  are  pre- 
sented to  the  drawee,  and  if  he  agrees  to  pay,  he  writes  the  word 
N  Accepted"  and  his  name  across  the  face  or  on  some  other  part 
of  the  bill,  and  returns  it  to  the  holder.  The  drawee  is  then  the 
Accepter,  and  responsible  for  the  payment  of  the  bill  when  due. 

287.  Explaiu  the  operations  of  Exchange.  28S.  When  are  some  bills  payable? 
Others?  289.  What  may  the  payee  do  with  a  bill?  What  is  the  buyer  called? 
The  owner?  How  does  the  6eller  of  a  bill  become  responsible  for  the  payment 
©fit?    What  is  the  maturity  of  a  bill?    200.    What  is  it  to  accept  a  bill? 


9111.    If  the  drawee  ieotinei  to  peg  or  aeeepl  the  bill,  the 

/bolder  em;  lii-tr  called   a    All  notice 

of  the  refusal  to  the  drmwot  and  each  Indofoor.    Thi 

999.    A  hill  should  bi  |  for  payment  doting  the 

is  hoars  of  the  day  oa  which  it  aiaiaff%and,  if  the 

accepter  fail <  10  make  pa_\  ment,    tin-    holder    should  j>ro(rs(  it  j'»r 

'itjintut  by  giving  the  proper   notice  to   the    drawer  and  the 

pal  indorsers.      It"  this  notice  il    n<»:    given    in    due    time   the 

indorser   eease  to  be  boldea  for  the  payment. 

9921.     The  United  States  annually  export  to  and  import  from 

>ods  to  the    value   of  hundreds   of  millions    of  dollars. 

1   the    imports,  and    sometime-  |he 

When  our  export*  to  a  given  country,  England, 

ur  imports  from  England,  the  balance  of  trade  is  in 
ourfa\or:  England  owes  US  more  than  we  owe  England,  and 
reliant  I  here  wi-h  to  sell  bills  drawn  on  England, 
for  the  purpose  of  eolleeting  their  duo  in  England,  than  wi>h  to 
DOJ  for  the  purpose  of  paying  their  debts  there,  and  consequent- 
ly, the  supply  being  greater  than  the  demand,  bills  on  England 
will  sell  at  a  discount.  When  the  balance  of  trade  i*  in  favor 
of  England,  OUT  indebtedness  is  greater  than  that  of  England, 
and  bills  on  England  will  sell  at  1  premium.  This  ehan. 
the  price  of  bills  is  called  the  COUESI  of  K.\i  iiam,i;  The 
variation  in  the  price  of  bills  can  never  be  very  great,  for  mer- 
chants will  not  pay  more  for  premium  than  the  cost  of  freight 
and  in-urnnee  to  transport  specie. 

991.     Bilk  of  exchange,  payable  after  sight,  like  promi- 

;  to  I  discount   for   the   term    of  credit,   the    di  — 
count  being  Computed  on  the  face  of  the  bill. 

993.     In  tl.  |  the  exchange  value  of  the  pound 

291.    Kor  what   is  a  bill  protested*    By  whom?    How?    29*2.    When  should 
a  bill  be  presented    for   payment?    What   is  necessary  to   hold   tin 
291.    When  is  the   balance  of  trade  in   our  favor?    When  agai. 
does  this  affect  the  price  of  bills  of  exchange?  What  is  the  Course  of  Exchange? 
MUttf  the  variation  be  great?     294.    Are  time  bills  subject  to  di-count? 
495.    What   u  the  exchange  value  of  the  X*     What  the  commercial  value? 
20 


230  PERCENTAGE. 

sterling  is  $4.44|  and  bills  of  exchange  are  drawn  upon  tin. 
basis,  but  the  intrinsic  and  commercial  value  is  about  9  per  cent, 
more  than  the  exchange  value ;  thus, 

Exchange  value  of  l£  =s  $4.44 % 
Add  9  per  cent,  as      .40 
Average  commercial  value  of  l£  =  $4.84$  ; 
\;.  when  exchange  on  England  sells  at  a  premium  of  9  per  cent, 
it  is  at  true  or  commercial  par. 

29G.  Problem  1.  To  find  the  cost  of  a  draft  or  bill 
of  exchange. 

Ex.  1.  $1000.  Boston,  June  4,  18G2. 

At  Bight,  j»:iy  John  Jones,  or  order,  one  thousand  dollars,  value 
received,  and  charge  the  same  to  my  account.  A.  Tyler. 

To  Mfcwa.  Smith  ft  Dana,) 
Merchants,  Chicago.  ) 

What  is  the  cost  of  the  above  draft  at  2  per  cent  discount  ? 

#1000  X  .98  =  $980,  Ans.  Since  exchange  is  at  2  per 

^  cent,  discount,  each  dollar 

costs  98  cents,  i.  e.  the  bill  costs  .98  (98  hundredths)  of  its  face. 

2.  $320.  l^ttsburg,  Aug.  G,  18G2. 

Sixty  days  after  sight,  pay  to  S.  Day,  or  bearer,  three  hun- 
dred and  twenty  dollars,  value  received,  and  charge  the  same  to 
the  account  of  T.  Fox  &  Co. 

To  Alfred  Stearns, ) 
N'  m  York.      ) 

What  is  the  cost  of  this  draft  at  3  per  cent  premium  ? 

OPERATION. 

$320 

9.G  0  =  premium  on  $320  at  3  per  cent 

3  2  9.6  0 

3.3  6  =  discount  on  $320  for  60  days  and  grace. 
$ 32~6\2  4  =  cost  of  draft,  Ans. 

3.  What  is  the  cost  of  a  draft  on  St.  Louis  for  $832'),  at  2 
per  cent,  discount  ? 

4.  What  is  the  cost  of  a  draft  on  New  York  for  $7850,  at  1 
per  cent  premium  ? 


n   1.     An  <  'Miiutry  whore   it    ia  <Ir:i\vn,   is 

called  a  dmfl  or  an  inland  bill  of  (.!■■  r  drawn  in  one  country 

and   payable  in  another,   is  called   ft  foreign  bill  of  exchange.     In  making 

i  bills  it  is  customary  to  draw  ■  let  of  two  or  more  bills  of  the  same 

tenor  and  date,  each  containing  a  clause,  in  parenthesis,  which  renders  all 

the  bills  in  the  set  worthless  except  the  one  first  presented  to  the  drawee. 

These  bills  arc  sent  in  different  vessels  so  that,  if  one  or  more  of  the  set 

r  loat  on  the  passage,  there  may  be  no  unnecessary  delay  in 

obtaining  the  money. 

l>000£.  Boston,  May  12,  1862. 

At  tight  of  this  first  of  exchange  (second  and  third  unpaid), 
pay  to  the  order  of  John  Flint,  in  London,  two  thou -and  pounds 
sterling  ed,  and  charge  the  same  to  my  account. 

David  Fay. 
To  Geom  Peabodv  &  Co.,) 
Jeers,  London.  j 

What  is  the  cost  of  this  bill  in  United  State9  money,  at  9^ 

at.  premium  ? 

OPERATION. 

S-l.W$X2000  =  $888  8.8  8{f  =  2000£. 

8  4  4.4  4  $  =  premium  at  0  \  per  cent. 

$  9  7  3  3.3  3  $  =  cost  of  bill,  Ans.     Hence, 

RULE.  First,  if  necessary,  find  the  value  of  the  bill,  at  par, 
in  United  States  money  ;  then  increase  or  diminish  this  value  as 
the  rale  of  exchange  and  the  term  of  credit  may  require. 

6.  Stuart,  Field,  &  Co.,  of  New  York,  bought  of  J.  &  P. 
Smith,  a  set  of  exchange,  payable  at  tight  for  800£,  on  Bates, 
Baring,  &  Co.,  London,  at  8|  per  cent,  premium.  What  was 
the  cost  in  U.  S.  money  ?  Ans.  $3866.  66$. 

nl     An  English  coin  worth  IX  is  called  a  sovereign. 

7.  I  wish  to  pay  a  debt  of  1200£  in  Liverpool.  Which  can 
:  afford,  to  buy  sovereigns  at  (4.85  and  pay  2  per  cent,  for 
il   and   insurance,  or  buy  a  set  of  exchange  at  9  J  per  cent. 

premium?  ^109.73^  by  buying  the  bills. 

SOO.     Rule  for  fluding  the  cost  of  a  bill  ?     What  is  an  inland  bill  !     A  for- 
eign bill?    A  sovereign? 


232  PERCENTAGE. 

297.  Problem  2.  To  find  the  face  of  a  bill  which  a 
given  sum  in  United  States  money  will  buy. 

Ex.  1.  When  exchange  n  at  0?  por  cent,  premium,  what  is 
the  face  of  a  bill  on  London  which  I  can  buy  for  $4990? 

1  £  =  $  4.4  4 1 ;  S  4.4  4  f  +  9 1  per  cent  =  S  4.«  7  £,  cost  of  1  £ ; 
$  4  3  9  0  -j-  §  4.8  7 1  =  9  0  0,  No.  pounds  in  bee  of  bill, 

2.  My  agent  in  Chicago,  bought  a  draft  on  New  York, 
per  cent  premium,  for  $8160  ;  what  was  the  face  of  the  draft  ? 

$1+2  per  cent  =  $  1.0  |  -  l. 

$  8  1  6  0  -f-  1.0  2  =  $  8  0  0  0,  Am.     I  i 

Rule.  Divide  the  cost  of  the  bill  by  the  cost  of  a  bill  for  $lt 
l£,  etc.,  and  the  quotient  will  be  the  face  of  the  bill  in  dollars, 
pounds,  etc. 

3.  A  Boston  merchant  bought  a  draft  on  Chicago,  at   9   per 

cent,  discount,  for  $.3820  ;  what  w;ts  the  face  of  the  draft  ? 

Ans.  $u000. 

4.  Bought  a  set  of  exchange  on  London,  at  9£  per  cent,  pre- 
mium, for  $4168.30  ;  what  debt  in  London  may  be  paid  by  this 
sum?  Ans.  856.5£  =  856£  10* 

EQUATION  OF  PAYMENTS. 

298.  Equation  of  ?  h  the  method  of  determin- 
ing when  several  debts  due  from  one  perm  to  another,  payable 
at  different  times  may  be  paid  at  one  time,  so  that  neither  party 
may  aafler  1<>— .    The  equated  fume  u  the  date  of  payment. 

The  time  to  elapse  before  a  d^bt  becomes  due  is  called  the 
term  of  credit.  The  average  term  of  credit  is  the  time  to  elapse 
before  the  equated  time, 

299.  PROBLEM  1.  To  find  the  equated  time  when 
all  the  terms  of  credit  begin  at  the  same  date. 

Ex.  1.  On  the  1st  of  Jan.  A  owes  B  S2,  payable  in  4  months 

207.    Rule  for  finding  the  face  of  a  bill?    29S.    What  is  Equation  of  Pay- 

Uiciits  ?    What  the  equated  time  ?     Term  of  credit  ?    Average  term  of  credit* 


EQUATION    U; 

ivable  in  8  month- ;  what  i>  the  average  term  vl' 
and  the  equated  lime  of  payment? 

in>D. 

4x2=    8  Tlic  privilege  of  ki 

8X6  =  48  ing  $2  for  4m.   is  the 

gv      7T  .-nine  Sfl  tin-  privilege  of 

'      —  for  8m. ;  so 

7m.,  1st  Ans.  ,,.  is  the  same 

Jan.  1  -u  7m.  =  Aug.  1,  2d  Ans.  |]   for  48m.  J  ...  for 

the  two  debts,  A  might 
hoep  Si  Tor  56m.,  but  as  he  has  $8  to  keep,  he  may  retain  it 
."»6m.,  viz.  7m.,  and  7m.  from  Jan  1,  extend  to  Aug.  1, 
the  equated  time.     Hence, 

BULB  1.  Multiply  each  debt  by  the  number  expressing  the  time 
to  elapse  before  it  becomes  due,  then  divide  the  sum  of  the  products 
by  the  sum  of  the  debts,  and  the  quotient  is  the  average  term  of 
credit  Add  the  average  term  of  credit  to  the  date  of  the  debts, 
and  the  result  is  the  equated  time. 

Remark.     Express  each  time  in  months,  or  else  each  in  days. 

8ECOND    METHOD. 

The  interest  of    $  2  for  4m.  =     4  c 
"  SG  "  8m.  =  2  4  c. 

Sum  of  debts  =  S  8  2  8  c.  =  total  interest. 

Now  the  question  is,  in  what  time  will  the  interest  on  the  sum 
of  the  debts  be  the  same  as  the  sum  of  the  interests  on  the  several 
debts?  This  may  be  found  l>y  dividing  the  total  interest  by  the 
interest  on  the  sum  of  the  debts  for  1  month;  thus,  interest  of 
S8  for  lm.  =  4c,  and  28c.  -~-  4c.  =  7,  number  of  months  fan  the 
ge  term  of  credit,  as  by  the  1st  method.     Hence, 

Krir  2.      Find  the  interest  on  each  debt  for  its  term  of  credit, 
then  divide  the  sum  of  these  interests  by  the  interest  on  the  sum  of 
!>(s  for  one  month,  and  the  quotient  will  be  the  average  term 
of  credit  in  months. 

id  the  equated  time  as  in  Rule  1. 

the  sam  of  the  debts  for  a  month,  it  ii 

299.  Rule  for  finding  average  term  of  credit  ?    Equated  time?    Second  method  T 
Explain  Ex.  1  by  each  method.    Second  Rule  ?    What  is  Note  1  1 
20# 


234  PERCENTAGE. 

only  necessary  to  move  the  decimal  point  two  places  to  the  left  and  divide 
by  2  (Art.  241  j,  for  the  interest  of  $1  is  just  half  a  cent  a  month. 

Note  2.  It  is  the  custom  of  business  men  to  consider  30  days  a  month ; 
also,  in  computing  interest,  to  neglect  the  cents  in  the  principal  if  they  arc 
less  than  50,  and  to  add  1  to  the  number  of  dollars  in  the  principal  if  the 
ctnt>  are  50  or  more. 

So  the  fraction  of  a  day,  in  equating  accounts,  is  neglected  if  less  than  k, 
and  it  is  counted  aa  1  if  it  is  i  or  more. 

Note  3.  Karh  method  above  given  is  much  used  by  accountants  in 
averaging  accounts,  but  the  second  is  thought  to  be  the  shorter  and  better 
method.  The  second  only  is  given  in  the  following  problems,  but  the  pupil 
will  practice  upon  either  or  both,  as  his  teacher  may  <L 

2.  July  6,  1861,  I  owe  to  John  Smith  $4.">,30,  payable  in  4m., 
75  id  8m.,  and  $3500  in  12m. ;  what  is  the  average  term  of 

credit  and  the  equated  time  ? 

Ill  Ads.  Average  term,  7.08  m.  =  7m.  2<».l. 
2d  Ana.  Equated  tin,  .  ft  b,  M,  1862. 

Note  4.  The  decimal  of  a  month  may  bfl  reduced  to  days  by  multiply- 
ing by  30  (Art.  176),  or  more  conveniently  by  taking  3  days  for  each  teuth 
and  1  day  for  each  3 }  hundredths  in  the  decimal. 

3.  $1500,  $2100  and  $2100  are  dM  in  4,  8,  and  12  months, 
respectively  ;  what  is  the  average  term  of  credit  ? 

300.  Problem  2.  To  find  the  equated  time  when  all 
the  terms  of  credit  are  of  equal  length,  but  begin  at 
different  times. 

In  solving  examples  where  the  terms  of  credit  are  equal,  it  is 
only  necessary  to  find  the  average  date  of  the  debts,  and  then  to 
this  date  add  the  term  of  credit. 

In  finding  the  average  date,  interest  may  be  computed  from 
the  date  of  the  first  bill,  or  from  any  other  date  /but  it  is  most 
convenient  to  compute  the  interest  from  the  frst  of  the  month  in 
which  the  first  bill  is  boitgld,  because  the  time  for  which  interest 
i<  to  be  computed  on  the  several  bills  i>  thereby  most  easily 
will  be  seen  by  the  following  examples. 

^J99.  What  is  Note  2  ?  Note  3  ?  Note  4  ?  300.  In  finding  the  average  date  of 
debts,  interest  may  be  reckoned  from  what  time  ?    Most  convenient  time  ?   Why  ? 


$300 

$0.60,  Int.  for  1 2d. 

-.     *     -  L8d 

3.6  0,    "    *  lm.  6d 

100 

2.4  0,     "     «  4m.  2 4 J. 

KgrATIU.N*    CM 

Tii  m  which  interest  on  uted, 

fa  Called  I 

.  1.  Required  the  equated  time  of  paying  the  following  bills 

of  gi  .  boaghl  on  a  credit  of  C  DXHltl 

0m    Mar.    1-J. 

Dm.    •■ 

lm.   Aj>r.      »',, 
4m.  July   -J  1. 

2 )  100)$  1200  $ 7.2  0,  total  inter 

lot  on  nun  of  bills  tin-  1  in.,  $6 

0-1-  6.0 0  =  1.2m.  =  Ira.  Gd. 
Thi>  gives  the  average  date  of  purchase  1  month  and  G  dajs 
from  Mar    I,  viz.  Apr.  <>.     To  this  add  the  term  of  credit,  6m., 
and  we  have  Oct  6   for  the  equated  time  of  pay m 

'.:.     Sinn-  the  time  for  which  interest  is  computed    tmdudte  Loth   the 
of  tin-  month  ami  the  day  of   purchase,  so  lm.  and  Cd.  from  M;ir.  1 
Lnl  :i-  ending  on  the  6th  of  Apr.  and  not  on  the  7th.     The  same 
principle  holds  in  the  following  examples. 

taxation.  The  time  for  interest  on  the  first  bill  is  0 
month-  and  12  days,  the  number  of  days  being  determined  by  the 
Date  OF  thk  BILL.  So  the  time  of  the  second  bill  is  Oin. 
:  of  the  third,  lm.  6d. ;  and  of  the  fourth,  4m.  2  Id. 
The  number  of  months  may  be  obtained  by  counting  from  the 
local  date  (e.  £•  for  the  fourth  bill  above,  April,  May,  Juno,  July, 
i.  o.  1,  2,  8,  4)  and,  for  conveniont  US6,  the  number  of  months 
/  the  l$fi  of  the  date  of  the  bills,  severally. 

reel  of  each   bill   is  computed   for   its  own    time   ami 
irriUen  at  the  right    The  aggregate  or  total  interest  on  the  bills 
(in  this  example,  $7.20)  is  then  divided   by  the  interest  of  the 
sum   of  the   bills   for    1    month    (|6),M    in  Ex.    1,  Art.  2'.' 
iii-th  in  the  average  date  of  purchase.     II< a 

Kt  tack   bill  from  the  first  Of 

month  in  which  ■'■'//  mm  bought  to  the  time  ef  the  pur- 

•  f  the  bills,  severally  ;  divide  the  sum   of  these   interests  by 

300.    Focal  date,  what  is  if  L    Number  of 

month.*,  how  found  !    Where  set!    Rule  for  finding  average  date  ?    Bqnttod 


236 


l'EK- 


the  interest  on  the  sum  of  the  bills  for  one  month,  ctn<?  the  quotieni 
will  be  the  number  of  months  from  the  focal  date  to  the  average 
date  of  purchase.  To  this  average  date  of  purchase  add  the 
term  of  credit,  and  the  equated  time  of  payment  is  found. 

2.  Required  the  equated  time  of  paying  the  following  bills, 
each  bought  on  8  months'  credit  ? 


Om. 
lm. 
3m. 

4m. 

June    9, 
July  15, 
Sept.  11, 
Oct.    10, 

1862,      $180 

8  4 
"             2  40 

9  6 

$  0.2  7,  Int.  for  9d. 

.6  3      "     "     lm.  15d. 
4.1  6      "     "     3m.  14d. 

2.0  8      "     "     4m.  K),i. 

2)1  0  0)  $600 

$  7.1  4,  total  inter 

3) 

7.14 

2.3  8m.  =  2m.  lid. 

.•.  Average  date  of  purchase,  Aug.  1  1.  1862. 

Equated  time  of  payment,  Apr.  11,  1863,  Ans 

3.  Bought  the  following  bill* 

on  6  months'  credit : 


May   12,  1862,  $100 

June    4,  "  150 

Aug.    6,      "  80 

24,      "  170 

What  is  the  average  date 
of  purchase  and  equated  time 
of  payment  ? 

1st  Ans.  July  5,  1862; 

2d  Ans.  Jan.  5,  1863. 

5.  Bought  the  following  bills 
on  6  months  : 

Jan.  8,  S :  9 

"  24,  20 

Apr.  18,  1200 

June  6,  4000 

"What  is  the  average  date 
of  purchase  and  the  equated 
time?        1st  Ans.  May  24  ; 

2d  Ans.  Nov.  24. 


4.  Bought  the  following  bills 
on  4  month- : 

Feb.  17,  1862,  $1200 

Mar.  25,       "  472 

u  30,       " 

July  21,       "  500 

What  is  the  average  date 
of  purchase  and  equated  time 
of  payment  ? 

Lsl  Ana.  Apr.  1,  18C2; 

2d  Ans.  Aug.  1,  1862. 

6.  Bought  the  following  bills 


on  6  mon*. 

Jan. 
u 

Apr. 
June 

8, 
24, 
18, 

6, 

$4000 

1200 

20 

12 

What 

is  the   average  date 

of  purchase  and 

time?        1st  Ans. 

the  equated 
Jan.    12; 

2d  Ans. 

July  12. 

EOIWTION    OF    I\\Mi:\ 

makk.     The    two   foregoing   example,  consisting  of  the 
bills,  with   the   order  "t'  purchf  led,  show 

clearly  that  thr  average  date  <>f  porcbaM   (and  consequently 
[oated  time  of  pajment)  i>  greatly  changed  by  buying  the 

smaller  bills  at  the  earlier  or  at  the  later  dates. 

301.  Problem  3.  To  find  the  equated  time  when 
the  terms  of  credit  are  unequal  and  begin  at  different 

lli.'  maturity  of  a  note  or  bill  is  the  time  when  it  becomes  due. 
The  process  for  finding  the  equated  time  of  payment  in  this 
Problem  is  the  same  as  for  finding  the  average  date  of  purchase 
in  Problem  2,  except  that  the  interest  is  computed  to  the  maturity 
of  the  bills  severally,  rather  than  to  the  time  of  purchase. 
Hence  no  new  rule  is  needed. 

1.   Required  the  equated  time  of  paying  the  following 
bills  of  goods? 

Cr.  Bills.  Int. 

Om.  Feb.   12,  4m.  $2  0  0        $4.4  0  for  4m.  12d. 

2m.  Apr.  15,  6m.  40  0  17.0  0    "   8m.  15d. 

4 in.  June     8,  2m.  3_00  9.4  0    "    6m.    8d. 

2)10  0)$9  0  0        $  3  0.8  0,  total  interest 
Int.  on  sum  of  bills  for  lm.  =  $  4.5  0 
30.80  -f-  4.50  =  6.84m.  =  6m.  25d.,  the  time  from  Feb.  1  to 
the  average  date  of  maturity,  i.  e.  to  the  equated  time.     Now  6m. 
25d.  from  Feb.  1,  1862,  gives  Aug.  25,  1862,  Ans. 

Explanation.  The  maturity  of  the  1st  bill  is  4  months  and 
1 2  days  from  Feb.  1  ;  the  maturity  of  the  2d  bill  (found  by 
adding  its  term  of  credit,  6m.,  to  the  2m.  15d.  from  the  focal 
date,  Feb.  1,  to  the  time  of  purchase,  Apr.  15)  is  8m.  15d. ; 
the  maturity  of  the  3d  bill,  found  in  like  manner,  is  6m.  8d. 

2.  Required  the  average  maturity  of  the  following  bills? 

Jan.  8ra.         $2000 

Feb.  21,         6ra.         3000 
June  6,         2m.  600 

300.  What  is  the  Remark?  301.  What  is  the  maturity  of  a  note  or  Wllf 
How  doe*  Problem  3  differ  from  Troblem  2'     Explain  Ex    1. 


233 

303.     PBOBLKM  4.     To    find    the    equated    timo    for 
paying  the   balance  of   an  account  which  has  both 
ami  credit  entries. 

Ex.  1.  From  the  accounts  of  A  and  B  it  appears  that 

A  owes  B  And  thai  B  owes  A 

$254,  due  July  18,  $500,  due  Aug.  15, 

475,    "    Sept.    6,  288, 

425,    "      "18,  012, 

46,    "    Oct.      9,  400, 

When  shall  P>  pay  the  balance  of  $G00? 

OPERATION. 

A's  Debt*. 

$24  I  I  for  18d. 

1  7  5  5.2  2  0    "    2m.  64 

425  InklM 

4  6  .7  ."»  9    "    3m.  M. 


u 

"      30, 

a 

Oct.    3, 

H 

"      21, 

0m. 

July  18, 

2m. 

Sept 

to. 

"     18, 

3m. 

Oct. 

Im.     Aug.  15, 
lm.        "      30, 

;;m.    Oct 

3m.       ••       21, 

B's  Debts. 
$500 

8 1  a 

400 

Sum  of  B's  debts = 

=$1800 

Sum  of  A's  debts  =$1200  $  1  2.2  7  1 ,  Total  interest 

on  A's  debts  from  tin'  focal  >lute.  July  1,  to  maturity,  i.  o.  the  in- 
terest that  B  would  ^rain  it*  A  paid  the  sum  of  his  debts,  $1200, 
on  the  1st  of  Jul  v. 

Tut. 
.7  5      for  lm.  1  "m1. 

2.8  8       "    2m. 

7.4  0      "  8m.  Sid 

$  2  3.5  1  6,  Total  interest 
on  B's  debts  from  the  focal  dot*\  July  I,  to  maturity,  i.e.  the 
tajteresj  A  would  gain  it"  B  paid  the  BOB  <>f  his  debts,  July  1. 

From  tl  e   it  ippeatt   that   if  each  party  paid  his  debts 

July  1,  A  would  gain  $23,516,  and  B  would  gain  SI  2.271  :  ,\ 
A^  net  gain  and  B's  net  loss  would  be  $23,516  —  SI  2.271 
=  $11,245.  Now  as  it  is  proposed  to  settle  by  B's  paying  the 
balance  of  the  account,  viz.  $600,  it  is  plain  he  may  keep  the 
I  after  July  1,  until  its  interest  >hall  equal  $11,245,  the  loss 
he  would  sustain  by  paying  July  1.  The  interest  of  $600  for  lm. 
is  $3,  and  $11,245  -t.  $3  =  3.748,  the  time  in  months.  Now 
3.7  18m.  =3m.  22d.  ;  .-.  the  time  of  payment  is  3m.  22<L  after 
July  i,  viz.  Oct   22,  Ans. 


EQl'-VTION    01  NTS.  239 

2.    The  BOOOmitfl  of  A  ami  B  .-how  that 

A  Offil  \\  And  that  B  01PM  A 

I.     due     Jan.    12,                                  i'S  due     Feb,     :», 

■•.  ■       Apr.     :», 

M  ij    1J,                               ifi  I,  »       July    18, 

140,              Jane    3,                            240,  "       Aug.  is, 

When  -hall  A  pay  the  balance  of  $100? 

OPERATIOX. 

A' s  Debts.  Int.  IT s  Debts.  Int. 

Om.  Jan.  12,  $624,  $1,248  Im.  Feb.    9,  $8  4  6,  $  2.2  1  9 

2m.  Mar.    6,     8  9  6)        0.8  5  6  3m.  Apr.    0,    9  60,  15.8  4 

4m.  Mav  1  2.     7  9  1,  1  C.l  4  8  Cm.  July  18,    4  5  4,  1  4.9  8  2 

.mi.  June   3,     14  6,  _3.7  2  3  7m.  Aug.  18,    2  4  0,        9.1  2 

$  2  4  0  0,  $  3  0.9  7  5  $  2  0  0  0,  $  4  2.1  9  1 

$2400  $42491 

00  3  0.9J_5 

2)1  0  0  )$400,  Bal.  of  account.  $  1  1.2  1  6,  Bal.  of  interest. 

Int.  for  lm.$£00)$l  1.2  1  6 

Time  in  m.  =  5.6  0  8  =  5m.  l.Sd.,  which,  reckoned  back 
from  Jan.  1,  gives  July  13  of  the  preceding  year  for  the  time  of 
settlement,  An-. 

Explanation-.  By  a  process  like  that  in  Ex.  1,  it  is  shown 
that  jf  A  and  B  each  paid  his  debts,  i.  e.  if  A  paid  the  balmier 
of  $400,  at  the  focal  date,  Jan.  1,  A  would  pain  and  B  would 
.1  91  _  $30,975=  $11,216;  /.,  evidently,  A  should 
pay  the  S 400  long  moogfa  brforr  Jan.  1,  so  that  its  interest  shall 
equal  $1 1.216,  the  pain  be  would  have  by  paying  Jan.  1.  This 
time  U  found  to  be  5m.  18d„  which,  reckoned  back  from  Jan.  1, 
gives  July  13  of  the  preceding  year  for  the  equated  time  of 
ment     Hence, 

To  equate  accounts, 

R\  :>utc  the  interest  of  each  item  of  the  account  from 

rat  date  to  its   maturity ;  find  the  sum  of  the  interests  on 

the  debit  items,  also  the  sum  on  the  credit  items,  and  subtract  the 

less  sum  from   the  greater;  divide  this  difference  by  the  interest 


304.    Explain  Ex   1.     I  J     Kule  for  equating  account*  which  hara 

both  debit  and  orcriit  items » 


240 


on  the  balance  of  the  account  for  one  month,  and  the  quo- 
will  he  the  tune  in  m  "een  the  focal  date  and  the 

tied  time   of  settlement,  the   time  to    be  reckoned   forward 
l  the  greater  interest  arises  on  the  greater  side  of  the  account, 
and  backward,  excluding  the  focal  date,  when  the  greater  inter- 
est arises  on  the  smaller  side. 

Note  1.  When  the  hunger  interest  arises  on  the  smaller  side  of  tlie 
account,  as  in  Ex.  2,  the  rule  may  require  the  settlement  to  be  made  before 
some  of  the  transactions  have  occurred,  a  result  which  is  obviously  impractica- 
ble, and  usually  some  other  time  of  settlement  is  more  convenient  than  the 
equated  time.  If  the  settlement  is  made  before  the  equated  time,  a  discount 
should  be  made ;  if  after,  the  interest  should  be  added. 

Ex.  3.  When  ought  A  to  pay  the  balance  of  the  following 
account,  and  for  what  sum  may  he  settle  June  6,  1863  ? 

Dr.  A  in  account  with  B.  (  r. 


1 

$ 

A]>: 

To  Mdse.,  6m 

356 

By  Mdse.,  4m. 

530 

June  18 

Mdse,,  4ra. 

875 

27 

Mdse.,  6m. 

651 

July      3 

*    Mdse.,  6m. 

433 

Julv    15 

"     Cash, 

300 

Ufl  Am.  .June  G,  186  1  ;  2d  Ans.  $171.08.   (See  Art.  253b.). 

Note  2.     In  I  1  is  the  most  convenient  focal  date,  the  earliest 

entry  l>cing  made  Feb.  6.  The  meaning  of  the  account  is,  that  A 
three  different  times,  bought  merchandise  of  B  to  the  amount  of  $356,  $875, 
and  $433,  severally,  the  1st  and  3d  bills  on  a  credit  of  6m.,  and  the  2d  on 
4m. ;  also,  that  on  the  6th  of  Feb.  A  sold  B  merchandise  worth  $530  on  a 
credit  of  4m.,  on  the  27th  of  May  merchandise  worth  $652  on  6m.,  and  on 
the  15th  of  July  lie  paid  B  $300  in  cash. 

4.  Required  the  equated  time  of  settling  the  following  account, 
and  the  sum  due  Oct  4,  1862  ? 


Dr. 

A  in  account  with  B. 

Ok 

1862. 

$ 

1862. 

$ 

Mar.    14 

To  Mdse.,  4m. 

452 

April  13 

Br  Cash, 

500 

May     8 

«   C 

1224 

May   21 

4m. 

1000 

•«       20 

'•Msc.,  8m. 

150 

Aug.  18 

■    Cash, 

192 

"      27 

•    Mi-  ,  6m. 

2496 

Sept.  11 

"    Cash, 

5420 

June   19 

M  t-e.,  3m. 

5724 

July    30 

Gin. 

88  I 

1st  Ans.  Nov.  4,  1862  ;  2d  Ans.  $3006.89. 


EQUATION   OF   PAYMENTS. 


241 


Note  3.  Not  unfrequently  a  business  man,  in  full  or  partial  payment  of 
.  £ives  his  note,  payable  ba  ft  givan  time  without  interest.  The  holder 
of  the  note  may  IndoffM  it  and  gal  it  diacoonted  (See  Art*  253b.),  thus 
obtaining  money  for  his  own  use  before  the  note  matures  ;  or  he  may  pass 
it  to  his  creditor  in  payment  of  his  own   i  a  noto  may  be  entered 

in  tin  account,  as  in  Ex.  4,  and  treated  in  the  same  way  as  merchandise 
bought  or  sold  on  credit. 

5.  When  was  the  equated  time  of  settling  the  following 
account,  and  what  was  due  Nov.  13,  1862  ? 

Or. 


Dr. 


A  in  account  with  B. 


1861. 

$ 

1861. 

S 

Nov.   18 

To  Mdse.,  4m. 

800 

Sept.  27 

By  lidaat, 

1200 

1862. 

Dec.    12 

"    Mdse.,  4m. 

800 

April    6 

"   Mdse.,  2m. 

350 

1862 

"      30 

|  ash, 

125 

May    15 

"  Mdse.,  4m. 

850 

May    15 

"    Note,    4m. 

1200 

July    18 

"  Mdse., 

625 

Oct.     12 

"    Mdse.,2ra. 

200 

IM  Ans.  Apr.  25,  1861  ;  2d  Ans.  $874.40. 

6.  "When  is  the  equated  time  of  settling  the  following  account, 
each  item  being  due  at  date,  and  what  shall  A  pay  on  the  27th 
of  July,  1862  ? 

Dr.  A  in  account  with  B.  Or. 


1861. 

$ 

Int. 

Om.  June  20 

ToM 

986 

3.2871 

5m.  Nov.  16 

"  Mdse., 

152 

4.205 

1862. 

8m.  Feb,  26 

"  Mdse., 

110 

4877 

124S 

12.360 

1861. 

lm.   July    4 
6m.   Dec  18 

1862. 
9m.   Mar.    5 


By  Mdse., 
"   Note, 

"   Mdse., 


I 

IN 

228 


Int. 

0.895 

7.524 


450  20  625 


836  29  044 


$  l  i  1  8 

836 


2  9.0  4  4 
1  2.3  6  9 


2)100)412,  Balance  of  acct.  1  6.6  7  5,  Balance  of  int. 

2.0  6  )  1  6.6  7  5  (  8.0  9  m.  =  8m.  3d. 
June  1,  1861  — 8m.  3d.  =  Sept.  27,  1860,  Iff  Ans. 
$  4  1  2  +  $  4  5.3  2  (Int.  for  1  yr.  lOni.)  =  $  4  5  7.3  2,  2d  An* 

7.  What  would  be  the  equated  time  of  settlement  in  Ex.  6, 
if  each  Ktem  were  on  a  credit  of  6  months  ? 


303.    What  is  Note  1' 
21 


Note  2r    Note  8f 


-1-  PERCENTAGE. 

Proof.  Some  of  the  debts  are  due  before  the  equated  time, 
and  some  after.  The  sum  of  the  interests  on  the  former,  from 
t/ieir  several  maturities  to  the  equated  tune,  will  be  equal  to  the  sum 
of  the  interests  on  the  latter  from  the  equated  time  to  their  several 
maturities.  When  the  account  has  both  debit  and  credit  itnn*, 
equate  each  side  of  the  account,  and  the  interest  on  the  two  sides 
for  the  time  between  the  respective  average  dates,  and  the  equated 
time  will  be  the  same,  or  nearly  the  same  (Art.  299,  Note  2). 

PROFIT  AND  LOSS. 

303.  "  Profit  and  Loss,"  as  a  commercial  term,  signifies 
the  gain  or  loss  in  business  transactions.  The  rule  may  refer  to 
the  ((hsolute  gain  or  loss,  or  to  the  percentage  of  gain  or  loss,  on 
the  purchase  price  of  the  property  considered. 

304.  Problem  1.  To  find  the  absolute  gain  or  loss 
on  a  quantity  of  goods  sold  at  retail,  the  purchase  price 
of  the  whole  quantity  being  given  : 

Rule.  Find  the  whole  sum  received  for  (lie  goods,  and  the  dif- 
ference between  this  and  the  purchase  price  will  be  the  gain  or  loss. 

Ex.  1.  Bought  1 6  bbl.  of  flour  for  $100  and  sold  it  at  $7  per 
bbl. ;  did  I  gain  or  lose  ?     How  mueh,  total  and  per  bbl.  ? 

2.  Bought  24  bbl.  of  flour  for  $168  and  sold  $  of  it  at  $6.75 
ami  the  remainder  at  $7.50  per  bbl. ;  did  I  gain  or  loss  ?  II<>w 
much  ?  Ans.   Gained  $6. 

3.  Bought  3cwt.  2qr.  181b.  of  sugar  for  $36.80  and  sold  it  at 
8Jc.  per  lb/;  did  I  gain  or  lose  ?     How  much,  total  and  per  lb.  ? 

4.  Bought  164yd.  of  broadcloth  and  287yd.  of  cassimere  for 
$1107;  sold  the*  broadcloth  at  $3  and  the  cassimere  at  $2.25 
per  yd. ;  did  I  gain  or  lose  ?     How  much  ? 

305.  Problem  2.  To  find  the  per  cent,  of  gain  or 
loss  when  the  cost  and  selling  price  are  given  : 

30  2.    Froof  of  rule  for  equation  of  payments?    S03.   What  is  Profit  and  Lo«i? 
To  -what  may  it  refer?    304.    Rule  for  finding  absolute  gain  or  loss? 


i  243 

El    1.    Booghl  4bbl.  of  flour  for  $32  and  sold  it  at  $9.50  pet 
bbl. ;  did  I  gain  or  lose?    How  much  percent? 

$9-3  0,  telling  prl  The  gain,  $0,  is  3«*  =  ^ 

4  of  tfae    whole    cost,  and    -j^ 


$3  8.0  0,  whole  sum  rec\I.         n_,iu,-,,l  ,<,  B  dean*!  (Art 
|  8  .,  173),  givei  .18J  ;  i.e.  the 

gain  is  1S^   per  cent  of  tho 
$6,         whole  gain.  cost>     Q, 

,  r>  =.182,  Am. 

Hi  i.k.      I [nviiig  found  the  total  gain  or  loss  by  Problem   1, 
male  a   common  fraction    hy   icritimj   the  gain  or  loss  for   the 
rator  and  the  cost  of  the  article  for  the  denominator,  and 
then  reduce  this  fraction  to  a  decimal 

Bought  501b,  of  wool  for  $20  end  lold  it  at  34c.  per  lb.; 
did  I  gain  or  lose?     How  much  per  cent.? 

Ans.  Lost  15  per  cent. 

3.  Bought  a  ca>e  of  boots  at  S  1  per  pair  and  sold  them  a: 
what  per  cent,  was  gained? 

4.  Bought  boots  at  So  per  pair  and  sold  them  at  $4;  what 
Wit  was  lost? 

Booghl   goods   for  $2000,  and,  la  one  year,  sold  the  - 
2155,  out  of  which  paid  S'.1-")  for  storage,  etc. ;  how  much 
per  cent,  on  the  first  cost  was  lost? 

306.    Problem  3.    To  find  the  selling  price,  the  cost 
and  gain  or  loss  per  cent,  being  given. 

.  1.    Bought  goods  for  $400;  how  must  the  same  be  sold 
so  as  to  gain  2"»  per  cent 

$400 
.2  5 

2  0  0  0  This  is  the  same  as  finding  the 

g00  amount   of  a   sum   of   money   on 

interest  for  1  year  at  25  per  cent 


<>.0  0  =  gain.  (\r 

$4  0  0. 

$5  0  0.  Am 


305.    Rul«  for  finding  the  per  cent,  of  gain  or  low! 


244  i  age. 

2.  Bought  a  horse  for  $150,  but  it  being  injured,  I  am  willing 
to  lose  6  per  cent. ;  for  what  shall  I  sell  him } 

$  1  5  0  Tlii  :iihp  m  finding 

Am  preecm  worth  of  a  nm 

$100  =  loss.  <h,e  a  y^hence  discounting 

Z.  -^       «A       a,  j,     *  interest   (Art.  2o3b.). 

$150-$9  =  $i  LI, Am.     Bmmk  K 

Rule.  Multiply  the  purchase  price  by  the  per  cent,  to  be  gained 
or  lost,  written  decimally,  and  add  the  product  to,  or  subtract  it 
from,  the  purchase  price. 

Rimflll  a  farm  for  $4848;  for  what  shall  I  sell  the  sain 

grin  6  pat  eeaL?  *  $5090.40. 

4.  Bought  3cwt.  of  sugar  a;  m  shall   the  same  be 

sold  pec  lh.  M  M  to  gain  10  per  pent  '■ 

_rht  a  house  for  $3500,  expended  $750  in  repairing  it, 
and  then  sold  it  so  as  to  lose  15  per  cent,  on  the  whole  I 
what  did  1  rteetal  for  it? 

307.  Problem  4.  To  find  the  first  cost  of  an  article, 
fcta  letting  p  I  gain  or  loss  per  cent,  being  given. 

Ex.  1.  Bold  wheat  at  $1.50  per  bushel,  and  thereby  gaii 
I  Ifae  eo-t  ;   what  was  the  pfUVQaM  price? 
\%%  =  £  That   which    ■ 

4  of  $  1.5  0  ss  $  1.2  0,  Ans.        was  sold  for   I  J  .   .  .  .  the 

cost  was  }§8  ss  |  of  the 
•effing  price  ;  heme  the  cost  was  $  of  $1.50=  $1.20. 

2.  Sold  apples  at  $1.80  per  barrel,  and  (hereby  lost  10  per 
0Q  the  cost ;  what  was  the  cost? 

iftp.  =  if.  The  cost  was  Vo°  ==  V  °<*  the 

J,£of  $1.8  0  =  $2,  Ans.       selling  pri<  t  was  ty 

of  $1.80  =  $2.     Hence, 

BULK.  Mate  a  fraction  by  writing  100  for  a  numerator,  and 
100-J-  the  gain  per  cent.,  or  100 —  the  loss  per  cent.,  for  a 
denominator  ;  then  multiply  the  selling  price  by  this  fraction. 

300.  Rule  for  finding  the  selling  price,  the  cost  and  gain  or  loss  per  cent,  being 
given?  30T.  Rule  for  finding  the  first  cost,  the  selling  price  and  gain  or  losi 
ptr  cent,  being  given? 


PROFIT    AND    I  245 

3.  Sold  G  yard-  of  elota  for  I2&88,  and  gained  12  per  cent 
on  tbecosti  wlmt  irai  the  porchaM  price  per  yard  ?    Ana. 

Bold   l"  ihereaoftba   PHcfabarg  ft,  EL8tockfor  -1090, 

gainin  nt.  on  the  coat ;  what  did  I  pay  per  >1. 

llin-  95  Hi.  of  soger  tor  $2,  I  bsc  20  par  cent  oo  the 
;  what  was  the  cost  per  lb.  ? 

30S.   Proi  The  selling  price  of  goods,  and! 

the  gain  <>r  loss  per  cent,  being  given,  to  find  what  would 

I  pr  lost  per  cent,  if  sold  at  sorao  other  price. 

a  pair  of  oxen  for  $175  and  gained  5  per  cent. ; 

what   per  cent  should  I  have  gained  it   I   had  Bold  them  fo» 

. 

f9g  =  f  The    proposed    price    ia 

J  of  10  5  =  120  ?98  =  f  of  the  actual  Bell- 

i  2  0  — 10  0  =  20,  Ans.        log  price,  bot  the  actual  aelL. 

ing  price  is  1<>">  per  cent,  of 
the  cost,  .*.  the  proposed  price  is  $  of  1<»,">  per  cent.  =  120  per 
cent,  of  the  cost;  hence  120  per  cent. — 100  per  cent.  =  20 
per  cent,  would  be  the  gain  per  cent,  if  the  oxen  were  sold  for 

2.  Sold  a  farm  for  $5000,  and  thereby  made  25  per  cent. ; 
should  I  have  gained  or  lost,  and  how  much  per  cent.,  if  I  had 
sold  it  for  S3500  ? 

M$$  =  fft  =  ^;  A  of  125  =  87|;  100  —  87*  =  12*  = 
loss  per  cent.,  Ans. 

The  proposed  price  is  found  to  be  87*  per  cent,  of  the  cost, 
.-.  there  would  be  a  loss  of  12$  per  cent,  if  the  farm  were  sold 
B500, 

Rule.  Make  a  fraction  by  writing  the  proposed  piece  for  the 
numerator^  and  the  actual  price  for  the  denominator,  then  multiply 
tiki  per  cent,  at  which  the  article   is  sold  by  this  fraction,  and  the 

'd  will  be  the  per  cent,  at  the  proposed  price.      The  dijf'< 
between  the  product  and  100  is  the  gain  or  loss  per  cent,  at  the 
j)roposed  price. 

308.    Bute  for  finding  Ioh  or  gain  per  cent,  when  goods  are  sold  at  a  proposed 

priN  I 


246  PERCENTAGE. 

3.  Sold  flour  at  $7  per  bbl.  and  thereby  gained  If  per  cent.; 
what  per  cent,  should  I  have  gained  if  I  had  Bold  it  at  |7.85  ? 

Ana.  16  per  cent. 

4.  Sold  beef  at  $G  per  cut.,  and  thereby  lost  1  per  cent. ; 
should  I  have  gained  or  feet,  and  DOW  much  per  cent.,  had  I  sold 
it  at  16.50? 

5.  Sold  a  watch  for  $21,  and  gained  5  per  cent,  on  the  ej 
had   I  sold  it  for  $18  should  I  have  gained  or  lost,  and  how 
much  per  cent.  ? 

5109.    Problem  G.    To  mark  jroods  so  that  the  mer- 
chant  may  fall  a  certain   per  cent,  on   the  marked   p 
and  yet  sell  the  goods  at  cost,  or  at  a  certain  per  cent. 
above  or  below  cost. 

(a)  To  sell  at  cost 

Ex.  1.  II«>\\  shall  I  mark  a  coat  that  cost  me  $18  so  that  I 
may  fall  10  per  cent,  from  tlue  marked  price  and  yet  ieU  the  coat 
at  00  W  =  V  J  V  of  $18  =  WO,  Ana. 

Since  I  am  to  fall  10  per  cent,  it  follows  that  the  cost,  $1 
only  iVg  =  &  of  the  marked  price,  and  if  $18  is  -ft  then 
will  be  }  of  $18  =  $:?,  and  \\\  will  be  1,0  times  $2  =  J 
i.  e.  the  marked  price  will  !.«•  \;>  of  $18  =  $20,  An-. 

Proof.      10  per  cent,  of  $20  =  Si',  which  taken  from  $20 

leaves  $18,  the  cost.    Hen 

Bulb.  Makt  a  fraction  by  writing  1 00  for  the  numerator, 
and  100  diminished  by  the  )>er  cent,  to  be  abated  for  the  denomi- 
nator ;  multiply  the  cost  by  this  fraction,  and  the  product  will  be 
the  marked  price. 

2.  Bought  a  case  of  watches  at  $28.50;  at  what  price  shall  I 
mark  them  to  enable  me  to  abate  G  per  cent.,  and  yet  sell  them 
at  cost  ?  Ans.  $25. 

(b)  To  sell  at  a  certain  per  cent,  above  or  below 
cost  : 

309.    Rule  for  marking  jroods  so  as  to  fall  a  certain  per  cent,  and  yet  sell  at 
•est?    To  sell  at  a  given  per  cent,  above  or  below  cost  ? 


:  IT    AND    1.  -  17 

Bulb.  Fintjimd  latanttay  prim  hp  ftioMmS;  then  find 
the  marking  juice  }»j  Pt  <t.) 

la  a  piece  of  broadcloth  at  $0  per  yard,  l»ut  it  being 
httterrfl  I   am  wining  to  lose  20  percent,  on  the  cost;  bow 

shall    I    mark  it  M  that  I  may  fall  M  per  cent,  from  the  marked 

price  '■ 

$">   =  cost.  $">  —  $1  =  $4,  selling  \ 

.20 
$1.00  =  loss.  W  X  $4  =  $5.33 i,  marked  pH 

Paid   $1  a  pair  for  a  case  of  boots  ;  how  shall  I  mark  the 
90  that  1  may  tall  10  per  cent,  from    the   marked  price  and 
yet  make  \'2k  per  cent,  on  the  0 

Paid   $8  each  for  a  case  of  bonnets  ;  how  shall  I  mark  the 
10  that  I  may  fall  10  per  rent,    from   the  marked  price  and 
yet  make  5  per  cent,  on  the  cost? 

Miscellaneous  Examples  in  Profit  and  Loss. 

1.   Bought   75  pounds  of  tea  for  $37.50  and  sold  £  of  it   at 
18  rent-  per  j>ound  and  the  remainder  at  5C  cents  ;  did  I  gain  or 
How  much  ? 
8.   What    per  cent,  do  I  gain  if  I  buy  boots  at  $3  per  pair 
and  -(11  them  at  $8.87$? 

3.  Sold  floor  at    $7.50  per  barrel  and  lost  $\  per  cent  on  the 
fa  what  should  it  be  sold  to  gain  12]  per  cent.  ? 
Paid  S3  per  yard  for  a  piece  of  lace;  how  shall  I  mark 
iMie  to  enable  me  to  fall  1«)  percent,  from  the  marked  price 
and  yet  gain  SO  per  eent.  on  the  eost  ? 

liought  hats  at  S3  per  hat  and  sold  them  at  $2.50;  what 
at  on  the  coal  wai  lost? 

>old  a  watch  for  $42  and  lost  12*  per  cent,  on  the  c 
what  was  the  0 

7.  Sold  eloth  at  $9  per  yard  and  lost  10  per  cent.  ;  should  I 
have  gained  or  lost,  and  DOW  much  per  cent.,  if  I  had  received 
$2.12*? 

>!d  him  so  as  to  gain  12 
ut.  ;   what  did  I  receive  for  him? 


248  PERCENTAGE. 


PARTNERSHIP. 


310.  Partnership  is  the  association  of  two  or  more  per- 
sons in  business. 

The  company  thus  formed  is  called  a  firm  or  house. 

The  money  or  other  property  invested  is  called  the  capital  or 
stock  of  the  company. 

The  profits  and  losses  of  the  firm  are  divided  among  the  part- 
ners in  accordance  with  their  interest  ID  the  business. 

311.  Problem  1.  To  find  each  partner's  sdmre  of 
gain  or  loss  when  their  capital  is  employed  equal  tim 

Ex.  1  A  and  B  trade  in  company  ;  A  furnishes  $100  and 
B  $800.     They  gain  $300  ;  how  shall  they  share  the  gain  ? 

A  furnishes  -rVo°o  =  h  °?  tne  stock,  .*.  he  is  entitled  to  £  of  the 
gain,  viz.  $100.  For  alike  reason  B's  gain  is  §  of  $300  =  $200. 

Or  we  may  solve  the  question  as  follows : 

$300 -f-  $1200  =  .25;  i.e.   the  profits  =  25  per  cent,  of  the 
stock ;    .-.   A's   share  of  profits  =  $400  X  .25  =  $100 
B's  share  of  profits  =  $800  X  .25  =  $200 

Entire  profit  $300  Hence, 

RULE  1.  Multiply  the  total  gain  or  loss  by  each  partner's 
fractional  part  of  the  stock;  and  the  products  will  be  the  respective 
shares  of  gain  or  loss  ;  or, 

Rule  2.  Find  what  per  cent,  the  total  gain  or  loss  is  of  the 
whole  stock,  and  then  multiply  each  partner  s  stock  by  this  per 
cent,  written  decimally. 

319.  Proof.  The  sum  of  the  shares  of  gain  or  loss  must 
equal  the  total  gain  or  loss. 

2.  A,  B,  and  C  form  a  partnership ;  A  furnishes  $4000,  B 
$5000,  and  C  $6000.  They  gain  $3000  ;  how  shall  the  gain 
be  divided  ?  Ans.  A's,  $800  ;  B's,  $1000  ;  C's,  $1200. 

310.  What  is  Partnership?  What  is  the  company  called?  "What  is  the 
capital  or  stock?  How  are  the  profits  and  losses  divided  among  the  partners? 
311.    Rule  for  finding  the  shares  of  gain  or  loss?    Second  rule?    312.    Proof? 


PABnraraiR 

3.  Ha.l  the  Mm  b  Ex.  I  bit  $7.">o,  what  part  of  the  lose 

should  each  partner  su-tain  P      How  many  dollars? 

Yns.  A,  ft  ;  15,  J  ;  C,  J. 

4.  A,  B,  and  G  mirage  in  trade.  A  puts  in  $G000,  B 
$10000,  and  C  $8000.     They  gain  $4000;  what  is  each  part- 

-  -hare  ? 

k.     These  rules  arc  equally  applicable  to  distributing  the  property 
of  a  bankrupt,  and  many  other  similar  problems. 

5.  A  bankrupt  whose  property  is  worth  $5000  owes  A  $3000. 
B  $1500,  and  C  $3500;  to  what  fractional  part  of  the  property 
is  each  creditor  entitled?    To  how  many  dollars? 

&   Divide   $1500    between  A,  B,  and    C  so  that  A  shall 
•ve  $2  as  often  as  B  receives  $3  and  C  $5, 
7.   A.  B,  and  C   hire  a  pasture,  for  which  they  pay  $90;  A 
pastures  ;{  cow-,  B  ~>.  ami  C  7  ;  what  part  of  the  rent  shall  each 
I  low  many  dollars  ? 
8.  A  and   li  hire  a  pasture  for  $12;  A's  horse  was  in  the 
pasture  45}  weeks  and  B'l  7^  weeks;  what  rent  shall  each  pay  ? 
'.'.  A.  B.  ('.and  I)  freight  a  ship  to  Canton;  A  furnishes 
•  worth  of  the  cargo,  B  $5000,  C  $7000,  and  D  SI  1000. 
They  pun  $5200;  what  is  each  one's  share  of  the  gain? 

10.  A  and    B   form  a  partnership    with  a    joint  capital   of 
0,  of  which  A  famishes  |  in  cadi,  and  B,  for  his  share, 
farntshes  160  yards  of'  broadcloth.    They  lose  $300;  how  shall 
the  loss  he  divided?     What  is  the  price  of  B's  cloth  per  yard  ? 

313.     Problem  2.     To  find  each   partner's  share  of 

gain  or  loss  when  their  capital  is  employed  unequal  times. 

1.  A  and  B  trade  in  company;  A  puts  in  $300  for  8 
months,  and  B  $100  for  9  months.  They  gain  $800  ;  what  part 
of  the  gain  belongs  to  each  ?      How  many  dollar-  ? 

A's  $300  for  8m.  =  $2  1  00  for  lm. 
B's   $400  for  Dm.  =  $3  60  0  for  lm. 

00  0  for  lm. 

It  is,  .-.,  as  though  the  joint  stock  were  $G000  for  1  month, 

31J*.     What  i*tlit ■  N..:.  '     313.  Exy'iuii  Ex.  1. 


250  PERCENTAGE. 

of  which  A  put  in  $2400,  and  B  |9606 ;  hence  A  is  entitled  to 
f$gg  =  £  of  the  gain,  and  B  to  gggg  =  £  ;  i.  e.  A  is  entitled  to 
§  of  $800  =  $320,  and  B  to  f  of  $800  =  $480,  Ada.     Hence, 

RULE.  Multiply  each  man's  stock  by  the  time  it  is  continued 
in  trade,  and,  regarding  the  products  as  the  respective  shares  of 
*,tock,  and  the  sum  of  the  products  as  the  total  stocky  proceed 
as  in  Problem  1 . 

2.  A  and  B  engage  in  trade;  A  furnishes  $4000  for  12 
months,  and  B  $6000  for  11  months.  They  lose  $">70 ;  what  is 
the  loss  of  each  ?  Ans.  A"  -330. 

3.  A,  B,  and  C  engage  in  partnership;  A  furnishing  $600  for 
9m.,  B  $800  for  8m.,  and  C  $1000  for  12m.  They  gain  $1071 ; 
what  is  each  on  of  the  gain  ? 

4.  A,  B,  and  C  hire  a  pasture  for  $48.     A  pastures  3  h< 

for  8  weeks,  B  0  horses  for  G  weeks,  and  C  6  horses  for  7  weeks; 
whftl  part  of  the  rent  shall  each  pay? 

5.  B,  T,  and  C  enter  into  partner-hip,  doing  business  in  the 
name  and  ugnfttnre  of  B,  T,  and  C.     Jan.  1,  B  puts  in  $*5000,  T 

May  1.  15  puts  in  $2000  more,  ( 
and  T  take;  out  $1000.     Sept.  1,  B  takes  out  $3000,  T  poll  in 
•0,  and  0  S2O00.    At  the  end  of  the  year  tie  v  Mttfe,  having 
gained  $6400;  what  is  eaeh  partner's  share  of  the  gain? 

Asm,  Wi  ttO0O,Ti  £  »oo. 

6.  Jan.  1,  1860,  B  commenced  business  with  a  capital  of 
$3000.  Sept.  1,  1860,  wishing  to  enlarge  his  business,  he  took 
in  II  as  a  partner,  with  a  capital  of  $4000.  July  1,  1861,  they 
admit  L  into  the  partnership,  with  a  capital  of  $2500.  On  the 
1st  of  Jan.  1862,  they  dissolve  partnership,  having  gained 
$75,30 ;  what  is  each  one's  share  of  the  gain  ? 

7.  A,  B,  and  C  hire  a  pasture  for  $92.  A  pastures  6  horses 
for  8  weeks,  B  12  oxen  for  10  weeks,  and  C  50  cows  for  12 
weeks.  Now  if  5  cows  are  reckoned  as  3  oxen,  and  3  oxen  as 
2  horses,  what  part  of  the  rent  shall  each  pay?  How  many 
dollars  ? 

313.    Bute  for  finding  the  shared  of  gain  or  loto  when  the  capital  id  ii.  lb* 
unoqua.  timet? 


pARxanasHip.  ■-!.,[ 

ft.  A,  B.  and  C  hire  a  pa-turf  for  $.'500.     A  puts  in  10  oxen 

i,  and  96  sheep  for  2G  W( 

B  puts  in  7  oxen  for  24  '.-for  20  weeks,  and  66 

■beep  for  25  weeks ;  ('  poti  in  28  oxen  for  H  weeks  L2  cows  for 

12  weeks,  and  o."{  sheep  tor  10  weeks.     Now,  if  11  sheep  arc 

>ned  as  1  cow,  and  3  cows  as  2  oxen,  what  i<  the  ooel  per 

week   tor  a  Bheep  ?  a  cow  ?  an  ox?     How  many  dollars  does 

man  pay  for  sheep?  cows?  oxen?     What  part  <>f  (he  rent 

each  man  pay?   How  many  dollars? 

Ana.  Coal  per  week  for  a  sheep,  lcV;  a  cow,  10c;  an  ox, 

2le.      A   pay-   for  sheep,   $:57.41;   tor  COWS,   S.;3.f>0;   for  I 

B  pays  for  sheep,  $_>  1  ;  for  cows,  $38.40  ;  for  oxen,  S  10.82. 
('  pays  for  sheep,  S7.20  ;  for  cows,  $L'."».oi;  for  oxen;  $-18. 
A  pays  Iff  =  $1 19.04;  B,  $£f  =  $102.72  ;  C,  ,gi  =  $7 

!.  1  ox  and  S.  Low  enter  into  partnership.     January  1. 

D  $5000,  bat  Low  puts  in  nothing  until  May  1  ;  what  shall 
he  then  put  in  that  the  partners  may  he  entitled  to  equal  shares 
of  the  profits  at  the  close  of  the  year? 

10.  Jan.  1,  1853,  A,  B,  and  C  form  a  partnership  for  1  year, 
and  each  furnishes  $3000;  Mar.  1,  A  furnishes  $1000  more; 
June  1,  B  withdraws  $.300,  and  C  adds  $500;  Sept.  1,  A  with- 
draws S2000  and  C  $500,  and  B  adds  $1500.  Having  gained 
$4000,  at  the  close  of  the.  year  the  partnership  is  dissolved. 
What   i>  each  partner's  share  of  the  pain? 

1 1.  A,  1>,  and  C  traded  in  company.  A  at  first  put  in  $1000, 
B  $12no,  and  C  $1800;  in  three  months  A  pat  in  $500  more  and 

"0,  and  C  took  out  $100;  in  7  months  from  the  comnienc.  - 
Stent  of  beniaeSS,  A  withdrew  all  his  stock  but  $700,  B  put  ill 
as  much  as  he  at  first  put  in,  and  C  withdrew  \  as  much  as  A  at 
any  time  had  in  the  firm.  At  the  end  of  a  year  they  found  they 
had  pained  10  percent,  on  the  largest  total  stock  at  anyone  time 
in  trade.  What  is  the  total  pain?  What  fractional  part  shaO 
eaeh  have  ?     How  many  dollars? 

(  A's  part,  H?  =  $107.G3i?{. 
Ana.  Total  gain,  $U0.     ]  B's  part,  =$ 

(  C's  part,  =  $ 

Proof,  =  $ 


EXAMPLES   IX   A9AL1 


EXAMPLES    IN    ANALYSIS. 

313  a.     1.  If  6  barrels  of  flour  cost  $  12,  what  will  1 1  barrels 
cost? 

2.  Tf  §  of  a  cask  of  wine  co-t  $85,  what  will  7  casks  cost  ? 

3.  Twenty  is  £  of  what  number? 

4.  Fifty-one  is  j^g  of  what  number  ? 
.">.    Nin.-ty-five  is  }J  of  what  number? 

G.  If  £}  of  a  ton  of  hay  cost  Do  shillings,  what  will  a  ton 
cost  ? 

7.  If  g$  of  a  cask  of  oil  is  worth  $7  1,  what  is  the  value  of 

In? 
8  Sixty-four  is  g  of  how  many  times  12? 

9.  Seventy-two  is  §  of  how  many  times  4  ?    . 

10.  A  man  sold  a  watch  for  $o:j,  which  was  f  of  its  0 
what  was  its  cost? 

11.  A  pole  is  f  in  the  mud,  9  in  the  water,  and  G  feet  above 
water  J  what  i-  :h<-  Length  of  the  pole? 

12.  A  ship's  erew  have  provisions  sufficient  to  last  12  men  7 
months;  how  long  would  tiny  la-t  2  1  men? 

13.  A  can  build  85  rods  of  wall  in  .'):;  days,  but  B  can  build  0 
rials  while  A  build-  7  |  how  many  rods  can   B  build  in  44  d.. 

14.  f  of  28  is  ft  of  how  many  fifth 

15.  /T  of  4  1  i-  |  of  how  many  third-  of  15? 

16.  $  of  27  is  V  of  "ow  many  twelfths  of  60  ? 

17.  A  fox  has  30  rods  the  start  of  a  hound,  but  the  hound 
runs  27  rods  while  the  fox  runs  24  ;  how  many  rods  must  the 
hound  run  to  overtake  the  fox?  An-.  851. 

18.  A  hare  has  32  rods  the  start  of  a  hound,  but  the  hound 
rtmfl  12  rods  while  the  hare  runs  8  ;  how  many  rod-  will  the 
hare  run  before  the  hound  overtakes  him  ? 

19.  A  man  being  aaked  how  many  sheep  he  had,  replied  that 
if  he  had  a-  many  more,  h  as  many  more,  and  I]  iheep  he  should 
have  100;  how  many  had  he? 

20.  A  detachment  of  2000  soldiers  was  supplied  with  bread 
sufficient  for  12  Ptekf,  allowing  each  man  11  ounces  a  day,  but 


Ea  is. 

finding  1<>.*>  barn  1-.  containing  2<>nlh.  each,  wholly  spoiled,  how 
m.inv  ounce-  may  each    man    eat   daily,  that   the   remainder  may 

fast  tbem  IS 

21.  A  detachment  of  2000  having  j  of  their  bread 
spoiled,  were  pal  apoo  an  allowance  of  12  oc  each  per  day  for 
IS  weeks ;  what  ra  the  whole  weight  of  their  breads  good  and 

had.  and  how  much  WSS  Spoiled? 

22.  A  detachment  of  2000  soldi,  r-  baring  lost  1  0."»  barrel-  of 

tghing  2001b.  each,  wen-  Allowed  but  l2oz.  each  per 

day  lor  12  weeks)  hut  if  none  had  been  loot,  tiny  mighl  have 
had  1-1  oz.  daily  ;  what  WM  the  weight,  including  that  which  wa- 
it- t,  and   DOW  much  was  left  to  sub-i>t  on? 

A  detachment  of  2000  soldi. m,  having  lost  j  of  their 
bread,  had   each    12oz.  per  day  for  12   weeks;    what   Wat  the 

weight  of  their  bread,  including  the  part  lost,  end  how  much  per 

might  each  man  have  had,  had  none  been  lo-t  I 

21.  A  gentleman  left  bis  son  an  estate,  ]  of  which  he  ipent 

in  7  months,  and  J  of  the  remainder  in  8  months  more,  when  he 
had  only  $5000  remaining;  what  was  the  value  of  the  estate? 

•j."»  The  quick-step  in  marching  being  2  paces  of  28  inches 
each  per  second,  what  i-  the  rate  per  hour?  and  in  what  time 
will  a  detachment  of  soldiers  reach  a  place  GO  miles  distant, 
allowing  a  halt  of  1^  hour-? 

Two  men  and  a  boy  engage  to  reap  a  field  of  rye;  one 
of  the  men  can  reap  it  in  10  days,  the  other  in  12,  and  the  boy 
in  1.")  days.     In  how  many  days  can  the  three  together  reap  it? 

27.  A  merchant  bought  a  number  of  bales  of  hops,  each  bale 
containing  .,  at  tin-  rate  of  $3  for  111b.,  and  sold  them 
at  the  rate  of  $5  for  121b.,  and  gained  $248  ;  how  many  bales 
did  he  bn  Ans.  7. 

28.  Suppose  I  pay  8f{  cent-  per  bushel  for  carting  my  wheat 
to  mill,  the  miller  tUttS  A  ,(,r  gnodingi  it  takes  <1 .}  bushels  of 
Wheat  to  make  a  barn  1  of  Hour,  I  pay  80  cvnN  each   for  barrels 

%l\  per  barrel   for  carrying  the  flour  to  market,  where  my 
■    -ell-  i*.' i  barrel-  for  'it  of  which  he  tsJo 

ftrrel  for  i  - ;  what  do  I  receive  per  bushel  for  my 

Ans.  87 ^  cents. 
22 


KATI«>. 


RATIO. 

31 1.  Ratio  is  the  relation  of  one  quantity  to  another  of  the 
same  kind;  or,  it  is  the  quotient  which  arises  from  dividing  one 
quantity  by  another  of  the  same  kind. 

315.  Ratio  is  usually  indicated  by  two  dots ;  thus,  8 :  4 
( -x presses  the  ratio  of  8  to  4. 

The  two  quantities  compared  are  the  terms  of  the  ratio;  the 
faK  tem  being  the  antecedent,  the  second  the  consequent,  and 
the  two  terms  collectively,  a  couplet. 

310.  Most  math«-m:itieians  consider  the  antecedent  a  divi- 
dend, and  the  consequent  a  divisor  ; 

thus,     8:4=8-4  =  }=  2, 

and     3:  12  =  3  -4-  12  =  A=«i  ; 

\ 

but  others  take  the  antecedent  for  the  divisor,  and  the  consequent 

for  the  dividend; 

thus     B:    i   =4-^8=|=  \, 
and     3  :  12  =  12  -r-  3  =  V  =  4. 

re  1.     The  first  method  w  often  called  the  English  method,  and  the 
other  the  French;  but  there  appears  to  be  no  good  reason  for  such 
tinction. 

Note  2.     The  first  is  a  direct  ratio  ;  the  second  is  an  inverse  or 
ratio.     The  first  being  considered  the  more  simple  and  natural,  is  adopted 
in  this  work. 

317.     The  Antecedent  and  consequent  being  a  divi- 
dend and  divisor,  it  follows  that  any  change  in  the  AB 
CEDENT  causes  a  like  change  in  the  value  of  the  ratio,  and 
any  change  in  the  o  \T  causes  an  OPPOSITE  change 

in  the  value  of  the  ratio  (Art.  84,  85,  and  131).     Hence, 

1st.  Multiplying  the  antecedent  multiplies  the  ratio ;  and 
dividing  the  antecedent  divides  the  ratio  (Art.  83,  a  and  b). 

Hi,  What  is  Ratio?  315.  How  indicated?  What  are  the  term*?  The  1st? 
The  2d?  The  two  collectively?  316.  Which  term  is  divisor?  Is  the  custom 
uniform?  Which  method  is  here  taken?  Why?  What  is  a  direct  ratio?  An 
inverse  ratio?     317.    Explain  and  illustrate  Art.  317  fully. 


RATIO. 

2d.  Multiply in<j  the  consequent  diri,Irs  the  ratio;  and  dividing 
/isn/ui  >tt  )/iiilti/>lics  the  ratio  (Art.  85,  <•  and  <1). 

.  Malt iph/i in/  both  a  Mown!  by  the  same 

numbtr,  <>r  diridiny  botk  by  the  same  number,  does  not  affect  the 
ratio  (Art.  8  I,  ■  and  b). 

31S.    The  antecedent,  eoMoejieatj  and  ratio  are  so  related 

■h  other,  that,  if  either  two  of  them  be  given,  the  other  may 
md;  thus,  In  12:3=:  4,  we  have 

antecedent  +  consequent  =  ratio, 
antecedent  -~  ratio  =  consequent,  and 
consequent  X  ratio  =  antecedent. 

310.     When  there  is  but  one  antecedent  and  one  consequent 
the  ratio  is  said  to  be  simple ;  thus,  15:5  =  3,  is  a  simple  ratio. 

3£0.    When  the  corresponding  terms  of  two  or  more  simple 
-  ire  multiplied  together  the  resulting  ratio  is  said  to  bs 
coinj,ouitiI ;  thus,  by  multiplying  together  the  corresponding  terms 
of  the  simple  ratios, 


(s=»-yli=!r 


we  have  the  com- 


pound ratio,    4  8:4=1  2 or  48  0:  12  =  40. 

A  compound  ratio  is  always  equal  to  the  product  of  the  simple 
ratio-  of  which  it  is  compounded. 

Note.     A  compound  ratio  is  not  different  in  its  nctture  from  a  simple 
ratio,  bat  it  is  called  comjwund  merely  to  denote  its  origin.') 

1 .  Whet  is  the  ratio  of  20  to  4  ?  Ans.  20  :  4  =  5. 

l\  What  il  the  nitio  of  2  to  Ans.  2  :  9  =  $. 

3.  What  is  the  inverse  ratio  of  20  to  4  ?         Ans.    fa  =  £. 

4.  What  is  the  inverse  ratio  of  2  to  9  ? 

5.  What  is  the  ratio  comjKmnded  of  8  to  6  aud  9  to  2  ? 

6.  Which  is  the  greater,  the  ratio  of  9  to  7  or  of  19  to  14  ? 

7.  Which  il  the  greater,  the  ratio  of  5  to  4  or  of  15  to  13  ? 

318.    What  of  mtcced.  .  nt,  and  ratio?    310.  What  is  simple  ratio t 

ue!    Its  imtiii.-'     Why  oaJled  compound? 


PROPORTION. 


PROPORTION. 

3£l.     Proportion  is  an  equality  of  ratios. 

Two  ratio-,  and  .-.  4  terms,  are  required  to  form  a  proportion. 

322.     Proportion  is  indicated  bj  neflM  of  doll ;  A 

8  :  4  :  :  6  :  3, 
which  is  read,  8  is  to  1  as  ()  or,  as  8  is  to  4  so  is  6  i 

or  it  may  be  Indicated  t! 

which  is  mid,  the  ratio  of  8  to  4  equals  the  ratio  of  6  feO 

Any  4  numbers  are  in  proportion,  and  may  be  written  and  . 
in  like  manner,  if  the  quotient  of  tlie   1st  divided  by  the  :M  is 

equal  to  the  quotient  of  the  1  DJ  the  4 1 1 1 . 

393.  Tlie  [si  and    1th  terma  are  called  extremes,  and  tl. 
and  fljd,  means.     The  l-t  and  8d  are  the  antecedents  of  the  two 
ratios,  and  the  2d  and  4th  are  the  consequents.     The  prodnj 

mes  is  always  equal  to  the  product  of  tie-  mean- ;  thus, 
in  the  proportion  8  :  4  : :  6  :  8,  we  have  8  X  3  =  4  X  6. 

394.  Since  the  product  of  the  extreaica  hi  opaasl  bo  tba 
product  of  the  means,  any  one  term  may  be  found  when  the 
other  khn  en  ;  for  the  prodnd  of  th<-  extremes  dii 

by  either  mean  will  ghre  tfa    Otfcf  .  and  the  product  of  the 

means  divided  by  either  tber  extreme. 

Fill  the  blank  in  each  of  the  following  proportions : 

1.     8:    2::     :    3.  Ans.  ?-^-?  =  12. 

Ans.iX8=12> 


2. 

C:    9::8:      , 

3. 

4  !        : :  2  :    9, 

A. 

:  16  : :  7  :  1  1 

•M«  What  is  Proportion?  338.  How  indicated?  Proportion,  how  read? 
When  are  four  numbers  in  proportion?  343.  Whut  are  the  l«t  and  4th  taMM 
called?  2d  and  3d?  1st  and  3d?  2d  and  4th?  The  product  of  the  extremes 
equals  what?  344.  How  many  terms  mutt  be  given?  How  can  the  other 
bt?  found? 


4 

:  0  :  3 

1 

:  4  :  3 

B 

1 

:  8  :  6 

1 

:  4  :  8 

4 

:  6  :  8 

3 

:  8  :  4 

8 

:  3  :  4 

8IMPLB  257 

325.     It  follows  from  Art.  317,  that   if  the   tit  and  2d,  or 

U  :>n«l  1th,  or  1-t  ami  3.1.  or  2d  ami  1th.  or  all  tour  terms  of  a 
proportion  an-  multiplied  or  divided  l.v  the  MUM  number,  the 
rv.Miltin;:  numbers   will   he   in   proportion. 

3*-2<>.  It'  1  numbers  are  proportional  they  will  he  in  pro- 
portion  in   8   different   orders  ;  thus, 

(1)    (liven  8  : 

i    Alt. -matin-  (1)  8  : 

(3)    Inverting  (1)  4   : 

i    Alternating  (3)  4  : 

(."))    Invi-rtin'j  (  1)  tad  transposing  con pl<  t- 

l    Alternating  (.".)  3  : 

Inverting  (.".)  G  : 

(8)  Alternating  (7)  6  : 

may  be  written  in  16  other  orders,  but  none  of 
t*i<>m  will  Ik;  in  proportion. 

Jfc\57.  When  the  means  of  a  proportion  are  alike,  the  term 
repeated  i-  a  mean  proportional  between  the  other  two,  and  the 
la-t  term  is  a  third  proportional  to  the  1st  and  2d  ;  thin,  in 
4  :  G  : :  6  :  9,  6  is  a  mean  proportional  between  4  and  9,  and  9 
is  a  third  proportional  to  1  and  G. 

328*  A  mean  proportional  between  two  numbers  may  be 
found  by  multiplying  the  two  given  numbers  together,  and  then 
resolving  the  product  into  too  equal  factors  ;  thus,  the  mean  pro- 
portional between  2  and  8  is  4,  for  2  X  8  =  16  =  4  X  4;  .*. 
2  :  4  :  :  4  :  8. 

320.  A  third  proportional  to  two  numbers  may  be  found  by 
di riding  the  square  of  the  2d  by  the  1st.  The  third  proportional 
to  5  and  10  is  20 ;  for  10*  -+  5  =  20 ;  .%  5  :  10  :  :  10  :  20. 

SIMPLE    PROPORTION. 

330.     In  all  examples  in  Simple  Proportion  there  are  three 

325.   What  terms  may  be  multiplied  without  destroying  the  proportion?    What 
divided?    326.  Iu  how  man)  order*  may  four  proportional  numbers  be  in  \ 

In  how  many  not  in  proportion f  327.  Wtmt  is  a  mean  proportional?  A 
third  proportional?  328.  How  is  a  mean  proportional  found?  320.  A  third 
proportional' 

22. 


2.58  PROPORTION. 

numbers  given  to  find  a  fourth  ;  .-.  Proportion  is  often  called  the 
Rule  of  Three. 

Two  of  the  three  yieen  numbers  must  be  of  the  tame  kind,  and 
the  other  is  of  the  same  kind  at  the  answer. 

Ex.  1.   If  3  men  build  C  rods  of  wall  in  a  day,  how  many  r 
will  6  men  build  ? 

This  example  may  be  :maly/ed  as  follows  :  If  3  men  build  0 
rods,  1  man  will  build  $  of  6  rods,  i.e.  2  rods ;  and  if  one  man 
build  2  rods,  5  men  will  build  5  times  2  rods,  i.e.  10  rod.-.  Ant.  ; 
but  to  solve  it  by  proportion,  we  say.  that  ,'5  men  have  to  5  men 
th<-  same  ratio  that  the  given  number  of  rods  has  to  the  required 
number  of  rods  ;  thus, 

3  men  :  5  men  : :  6  rods:  required  number  of  rod-. 

Now,  since  the  means  and   1-t  .  nn  find  the 

2d  extreme  by  dividing  the  produet  of  tin-  Beam  by  the  given 

•  DM  (Art.  Ml'  1 )  :   thin. 
C  X  5  =  30  and  30  +  3  =  10,  Ans.  as  before.     II 

331.    To  solve  an  example  In  Simple  Proportion, 

Rnxn,      Write  that  given  number  which  is  of  the  I  as 

the  required  answer  for  ttie  third  term  ;  consider  whether  the 
nature  of  the  question  requires  the  answer  to  be  greater  or  less 
than  the  third  term  ;  if  greater,  write  the  greater  of  the  two  re- 
maining numbers  for  the  second  term  and  the  less  f>r  the  / 
but  if  less,  trrite  the  less  for  tlie  second  and  the  greater  for  the 
frst;  in  either  case,  divide  the  product  of  the  second  and  third 
tt  mis  by  the  frst,  and  the  quotient  will  be  tin  t,  rm  $<>>njht. 

Notk  1.  If  the  first  and  second  terms  are  in  different  denominations, 
they  should  be  reduced  to  the  same  before  statin g  ,on. 

\\  km  ark.  Everyone  who  intelligently  solve?  an  example  by 
proportion,  does,  in  effect,  solve  it  by  analysis  ;  but  the  teacher 
should  use  much  care  on  this  point,  since  the  scholar  learns  much 
faster  when  h  -  a  question  than  when  he  merely  follows 

330.  Of  what  kind  must  two  of  the  three  given  numbers  be?  What  th« 
Other!    331.    Ru!e  for  solving  an  example  in  proportion?    Note  1?    Remark? 


PORTION.  259 

a  rule.     Lei  the  following  examples   be   solved  by  analysis  ami 

by  pro|M>r:i<.n. 

2.  If  a  man  earn  $24  in  2  months  how  much  will  he  earn  in 

[>  nooi 

•  .  rm#  we  are  seeking  for  dol- 

lars, we  make  ^2  1  the  3d  term, 

o  \  9  !  I»  al,,l    then,    M    a    man    will    earn 

' '  more  in  9  month-  than   he  will 

in  2  months,  we  make  9  the  2d 

term  and  2  the  1st.    To  analyze 

th<>  above.  We   say.    If  a  man  earn  $24    in    2   mouths  then    in    1 

month  In*  will  earn  A  of  $24,  i.  e.  $12  ;  and  If  he  earn  $12  in  1 

month,  then  in  9  months  be  will  earn  '.)  times  $12,  i.e.  $108, 

3.   If  15  bosh,  of  Wheat  make  3bbl.  of  flour,  how  many  bush- 
wheat  will  be  required  to  make  7bbl.  of  flour?  Ans. 

•1.    It'   bObosh.  of  wheat  make  8bbL  of  flour,  how  many  bar- 
of  flour  will  75 bosh,  of  wheat  make?  Ans.  15. 

It'  a   man   ean    walk   7o   miles  in  3  days,  how  far  can    he 
walk  in  h  il  Ans.  2(M.»  miles. 

C.   If  a  man  travel  6  1  miles   in  2  days,  how  long   will  it  tak»- 
him  to  travel  K><>  miles?  Ans.  5  days. 

7     [fa  locomotive  run  30000  miles  in  13  weeks,  how  far,  at 
that  rate,  would  it  run  in  52  week-? 

I1Y    PROroHTI  BT    CANCELING. 

1*:52  r: 39000:  4th  term.  4 

I  x  52  =  2028000  ;                39000  x  H 
000-i.  18  =156000,  Ana,      —  =lo6000,  Ans. 

8.  If  20  men  perform  a  piece  of  work  in  8  days,  in  how  many 
lays  will  1  men  perform  the  same?  Ans.  40. 

[f  2  l  eordi  of  w 1  .  what  will  18  cords  cost. 

1".    If   $80  pay  for  5   cords  of  wood,  how  many  dollars  will 
pay  for  12  cords?  Ans.  72. 

11.  If  4  cords  of  wood  cost  $20,  how  many  cords  may  be 
bought  tor  -  Ans.  9. 

12.  If  6  bones  eat    -12   bu-hels  of  oats  in  5  weeks,  how  many 

besheli  will  ill  in  the  same  time? 

coal  when  l  t 


260 


PROPORTION. 


14.  In  how  many  day9  can  6  men  build  a  boott,  it'  10  m<  n 
can  build  it  in  72  days? 

15.  If  7211).  of  cheese  are  worth  as  much  u  .'5<»lb.  of  bu- 
how  many  pounds  of  cheese  will  pay  for  10 lb. of  bul 

1G.  How  many  tons  of  coal  can  !•  bought  for  $84,  win 
tons  cost  $18?  .11. 

1  7.  If  9  horses  eat  a  ton  of  hay  in  20  days,  how  many  hfl 
will  cat  a  ton  in  30  da\  An 

18.  How  many  tons  of  hay  will  f>  horses  eat  in  2.~> 
8  hot  --'"  ton-  in  the   MOM  til* 

19.  If  I  pay  2s.  8d.  per  week  f*<»r  |  vrhel  shall 
I  pay  for  pasturing  1  I  0 


2  :  11 


2s.8«L 

11 


2)  29s,  4d. 
Ans.  14s.  8d. 


or, 


2:11 


11 


2)352d.: 
Ana   176<L  =  l4a.  8d. 


20.  If  I  pay  2s.  8d.  for  pt  »w  many  cows  can 
tetared  the  hum  time  for  14a.  Bd.1 

21.  If  8  acres  of  land  cost75£  6s.  4d.,  how  many  seres  may 
bebou-iht  for  I31X  16s.  Id.? 

22.  If  14  acres  of  land  cost  13l£   16s.  Id.,  what  will  8  m 

If  |  of  a  ship  cost  $9875,  what  are  2  of  her  worth  ? 
24.  If  $  of  a  barrel  of  flour  cost  |  it  trill  6bbL  t 

It"  a  man  walk  192  miles  in  6  days  of  8  hours  each,  in 
fiuw  many  days  of  12  hours  each  will  he  walk  199  mi!' 

26.  Lent  a  friend  $400  for  6  months;  afterwards   he  lent  me 
$300.     How  long  may  I  keep  it  to  balam  >r  ? 

27.  llow  many   yard-  of  cloth  §  of  a  yard  wide  are  equal  to 
2<»  yard*  1 1  yard   v. 

.   if  when  Hour  is  worth  $9  per  bbl.,  a  penny  loaf  weigh.* 
4oz.,  what  will  it  weigh  when  flour  is  worth  $6  per  bbl.? 

29.  If  10  hi  10  bmheh  of  oats  in  3  weeks,  how  many 
BOabelfl  will  12  horses  eat  in  the  same  tim« ■? 

30.  Three  men  can  do  a  piece  of  work  in  12  days  ;  how  many 
men  must  be  added  to  the  number  to  do  the  same  in  4  d. 


UMFL1  m  'PORTION.  261 

81.  A  lUp*l  crew  of  12  men  has  food  for  24  days,  how  many 
mm  mu-t  !»<•  di-chargrd  that  it  may  last  12  days  tag 

AO  for  31b.  of  tea ;  what  should  I  pay  for  91b.? 
f  *  ship  cost  $8000,  wh:i-  i  of  her? 

\:  8M  per  BVt,  what  is  the  cost  of  62$  lb.? 
It'  i  steeple  180  feel  l'i_rli  casts  a  shadow  240  feet,  what 
is  the  length  of  the  shadow  cast  by  a  staff  3  feet  high,  at  the 
same  time? 

Note  2.  Since  each  of  the  three  terras  in  the  above  example  is  in  /bet, 
the  learner  may  be  uncertain  which  numl>cr  to  place  us  the  third  term  ;  but 
he  has  only  to  notice  that  he  is  required  to  find  the  length  of  a  shndnr,  .-. 
die  third  term  should  be  the  number  expressing  the  length  of  shadow  in  the 
given  examplo,  viz.  240ft. ;  thus, 

180  :  3  : :  240  :  4th  term  =  4  ft.,  Ans. 

36.  If  a  staff  3  feet  long  casts  a  shadow  4  feet,  what  is  the 
high!  of  a  steeple  which,  at  the  same  time,  casts  a  shadow 
840  feet  ?  Ans.  180  ft. 

37.  It'  ■  staff  3  feet  long  casts  a  shadow  4  feet,  how  long  is  the 
shadow  of  a  steeple  which  is  180  feet  high,  at  the  6ame  time  ? 

38.  If  a  steeple  180  feet  high  casts  a  shadow  240  feet,  what 
is  the  liight  of  a  staff  which,  at  the  same  time,  casts  a  shadow 
4  feel 

39.  The  interest  of  $300  for  lyr.  being  $18,  what  is  the 
interest  of  $850  for  the  same  time? 

40.  The  interest  of  $800  for  6m.  being  $24,  what  principal 
will  gain  $45  in  the  same  time  ? 

41.  If  a  man's  salary  amounts  to  $2700  in  3  years,  what  will 
it  amount  to  in  11  years  ? 

12.  If  a  man's  salary  amounts  to  $9900  in  11  years,  in  how 
many  yean  will  it  amount  to  $2700? 

If  12£  yards  of  silk  that  is  |  of  a  yard  wide  will  make 
a  dress,  how  many  yards  of  muslin  that  is  lg  yards  wide  will 
be  required  to  line  it? 

4  1.  If  J  of  an  acre  of  land  is  worth  $36.40,  what  is  the  value 
of  l-r»v^  acres,  at  the  same  price? 

If  6  nun  can  mow  12a.  3r.  16rd.  of  grass  in  2  day>,  l>y 
working  0  hours  per  day,  how  many  days  will  it  take  thciu  to 
do  the  same  if  they  work  only  4  hours  per  day  ? 


iNJ  PROPORTION. 

.  If  2bbl.  of  flour  are  worth  as  much  as  3  cords  of  wood, 
how  many  barrels  of  flour  will  pay  for  45  cords  of  wood? 

17.  A  bankrupt,  owing  $25000,  has  property  worth  $15000; 
how  much  will  he  pay  on  a  debt  of  $6 

48.  A  man,  owning  J  of  a  6hip,  sells  §  of  his  share  for 
$20000  ;  what  is  the  value  of  the  ship  ? 

l:».  A  and  15  hired  a  pasture  for  $45.90,  in  which  A  pastured 

11  oxen  and  B  19 ;  what  shall  each  pay  ? 

50.  If  13  men  perform  a  piece  of  work  in  45  days,  how  many 
in.  n  BMMl  be  added  to  perform  the  same  in  $  of  the  time  ? 

."»l.  If  tin-  int.  rest  on  $700  is  $42  in  one  year,  what  will  be 
the  interest  on  the  same  sum  for  3  J  years? 

52.  How  many  yards  of  paper  2  feet  in  width  will  paper  a 
room  that  i>  LS|  feel  bog  12  feel  wide,  and  '.♦  feel  bigM 

Ill  pay  $168  for  63  gallons  of  wine,  how  much  wi 
shall  I  add  that  I  may  nil  it  at  $2  per  gallon  without  loss? 

54.  A  certain  house  was  built  by  30  workmen  in  98  days, 
but,  being  burn<  .1,  it  is  required  to  rebuild  it  in  60  days;  how 
many  men  must  be  employ. 

55.  A  of  1500  men  has  provisions  for  12  months 
how   long  will   the  same  provi>ions   last  if  the   garri 
enforced  by  300  m 

56.  If  a  piece  of  land  20  rods  long  and  8  rods  wide  contains 
an  acre,  how  long  nm.-t  it  be  to  contain  the  same  when  it  is  but 
2  rod-  wide. 

57.  If  the  earth  revolves  366  times  in  365  days,  in  what  time 
does  it  revolve  on.  *,  m. 

58.  A  wall  which  was  to  be  built  24  feet  high  was  raised  8 
by  6  men  in  12  days;  how  many  men  must  be  employed  to 

build  the  remainder  of  the  wall  in  12  days  more  ? 

59.  A  wall  was  completed  by  12  men  in  12  days ;  how  many 
men  would  complete  the  same  in  4  days  ? 

If  a  man  perform  a  journey  in  6  days  when  the  daycare 

12  hours  long,  in  how  many  days  of  8  hours  each  will  he 
form  t'. 

61.  A  cistern  has  a  pipe  that  will  fill  it  in  6  hours;  how  many 
ime  use  will  fill  it  in  15  minutes? 


COMPOUND  noteBTioH. 

-tern  has  8  pipes;  the  fuvt  will  iill  it  in  A  hours  the 
second  in  I  hours,  ami  the  third  in  5  hours;  in  what  time  will 
they  together  till  the  cistern  ? 

63.   Paid  $8.50  for  71h.  of  tea  ;   what  should  I  pay  for  191b.? 

A  can  cut  a  field  of  gftJo  fa  A  and  B  can  cut  it 

in  8  days.      In  w  hat  time  can  B  do  the  saim 

It '2  horses  can  draw  a  load  of  16  tons  upon  a  railway, 
how  many  horses  will  be  required  to  draw  72  ton 

A  farm  was  sold  at  $25.50  per  acre,  amounting  to 
$1925.25;  how  many  acres  did  the  farm  contain  ? 

A  garrison  of  1000  men  have  14oz.  of  brood  ooefa  per 
day  for  120  days;  how  long  will  the  same  bread  last  them  if 
each  man  is  allowed  but  12oz.  per  day? 

68.  If  W  of  a  ship  eool  $26000,  what  is  }).  of  her  worth? 

69.  At  $27  per  cwt.,  what  is  the  cost  of  37  £lb.  ? 

70.  The  earth  moves  1 9  miles  per  second  in  her  orbit ;  how 
far  does  she  go  in  3m.  27sec. 

COMPOUND  PROPORTION. 

332.  Compound  Proportion  is  an  equality  of  two  ratios, 
one  of  which  is  compound  and  the  other  simple  ;  thus, 

3  •  12  ) 

If*    o  r  : :  ^  :  9,  is  a  compound  proportion ; 

and  48  :  24  : :  18  :  9,  is  the  same  reduced  to  a  simple  form. 
Note.     Tho  compound  ratio  may  consist  of  any  number  of  couplets. 

333.  Every  compound  proportion  may  be  reduced  to  a 
simple  form,  and,  moreover,  every  example  in  compound  propor- 
tion may  be  solved  by  means  of  two  or  more  simple  proportions. 

Ex.  1.  If  6  men  in  8  hours  thresh  30  bushels  of  wheat,  in  how 
Stay  hours  will  2  men  thresh  5  bushels? 

BT    8IMPLE     PROPORTION. 

2  :  "J  1,  and 

30  :  5  : :  24  :    4,  Ans. 

334.    What  is  Compound    Proportion?    333.    May  an  example  in  compound 
proportion  be  solved  by  simple  proportion?    Analyze  Ex.  1. 


PROPORTION. 

In  solving  this  question  by  simple  proportion,  we,  in  the  fir-t 
place,  disregard  the  amount  of  labor,  and  inquire  how  long  it 
will  take  2  men  to  do  as  much  as  6  men  in  8  hours.  Having 
found  24  hours  to  be  the  answer  to  this  question,  we  next  d 
gard  the  number  of  men*  and  inquire  how  long  it  will  take  to 
thresh  5  bushels  of  wheat  if  30  bushels  are  threshed  in  24  hours, 
and  thus  obtain  1  hours  the  true  answer  to  the  question. 

In  this  operation,  the  given  number  of  hours,  8,  is  first  multi- 
plied by  6  and  the  product  divided  by  2,  then  this  quotic 
multiplied  by  5  and  the  product  divided  by  30 ;  but  it  will  an- 
swer the  lame  purpose  to  multiply  the  8  by  the  product  of  the 
two  multipliers,  6  and  .r>,  Ihaa  divide  th«-  number  so  obtained  by 
die  product  of  the  two  divisors,  2  and  30 ;  thus, 

BT   COMPOUND   PROPORTION. 

30    !  _5  \  : :  8  :  4th  UnL  .    2  (|  multiplied  by 

g0     3  0  30  for  a  divisor,  and  the 

g  product  of  6  and  5  is  mul- 


60)240(4, 
240 


tipli. •<]  by  8  for  a  dividend. 


It  will  be  seen  that,  of  the  first  two  couplets,  i  on  !  e  £  >  °n« 

«itio  is  less  than  a  unit  and  the  other  greater  ;  but  there  is  no 
impropriety  in  this,  for  one  condition  of  the  question  requires  the 
answer  to  be  greater  than  the  3d  term,  and  the  other  condition 
requires  it  to  be  less.     Hence, 

331.  To  solve  questions  in  Compound  Proportion, 
lvii  i  .  Write  that  given  number  tchich  is  of  the  same  kind 
as  the  required  answer  for  the  3d  term  ;  take  any  two  of  the  re- 
maining terms  that  are  alike,  and,  considering  the  quMsSm 
as  depenpixg  ox  these  alone,  arrange  them  as  in  simple 
proportion;  arrange  each  pair  of  like  terms  by  the  same 
principles ;  and  then  multiply  the  continued  product  of  the  2d 
terms  by  the  3d  term,  and  divide  this  result  by  the  continued 
product  of  the  1st  terms  ;  the  quotient  will  be  the  term  sought. 

334.    Rule  for  compound  proportion? 


COMPOUND    PROPORTION. 

1'ho  work  may  often  be  much  ■bridged  by  canceling  factors  in 
I  ami  3d  terms,  with  lil.  K    to). 

Ex,  2.  It*  6  men  in  15  days  earn  $135,  bow  many  dollars  will 
i  earn  in  18  <!.' 
ten  :    '-1  men 
15  -lav- :  18  days 
Ox  18  X  l  •••*>-    21870  =  continued  product  of  2d  and  3d  terms. 
6  x  15  =  90  =  continued  product  of  Lat  terms. 

An-. 

,  -     -  .  [. 

2        27 


:  lth  tena 


I  :-.m 


i 

tt 

9  x  17  =  848,  Ana, 


9) 


IS        (**:     ,M.:,An, 


8.  U  i  men,  in  24  daya  of  0  hours  each,  build  a  wall  40ft. 

9ft.  high,  and  4ft.  thick,  in  how  many  days  of  C  hour* 
each  can  8  men  build  a  wall  60ft.  long,  12ft.  high,  nnd  5ft. 
thick  ?  Ans.  45. 


8  men       :    4  men 
6  hours     :     '.»  hours 

40  ft.  long  :  60  ft.  long 

9  ft.  hinh  :  IS  ft.  high 
4  ft.  thick  :     5  ft.  thick 


24  days : 


If  a  family  of  6  persons  spend  S000  in  8  months,  how 
many  dollar-  will  be  required  for  a  family  of  10  persons  in  14 
asontha?  Ans.  1750. 

5.  If  a  family  of  6  persons  spend  $G00  in  8  months,  how 
many  months  will  $17">0  sustain  a  family  of  1<>  p 

If  a  family  of  6  persona  spend  $000  in  8  months,  how  large 
a  family  may  be  -u-tained  1  1  months  for  $1750? 

7.  If  the  transportation  of  12  boxes  of  sugar,  each  weighing 
4cwt.,  40  miles,  cost  $8,  what  must  be  paid  for  carrying  40 
boxes,  weighing  8|cwt  each,  7-"»  miles?  Ans.  $48.75. 

8.  If  -1  mm  dig  a  trench  84  feet  long  in  2  \  day-,  how  many 
men  can  dig  a  treni  I  long  in  1  da;.  .  10. 

It"    1    men   dig   a   trench  Sift,  long  and  oft.  wide  in  .*'  da\  «, 

many  men  can  dig  a  trench  420ft.  long  and  3ft.  wide  in  4 

Ans.  9. 
23 


pROPORi : 

in.  If  2  men  dig  ■  trench  SOIL  long,  5ft.  wide.and  3ft.  deep 
in  S\  days,  how  many  men  can  dig  a  trench  800ft  long,  2.\ft. 
wide. and  4ft.  deep  in  7  day-?  An-.   L 

11.  If  6  DMA  « I i lt  a  trench  of  4  degrees  of  hardness,  30ft. 
loOA  5ft.  Wide,  end  5ft.   deep   in  5  days,    how  many  men  can  dig 

ocfa  of  6  degrees  of  hardness  105ft,  long,  4ft.  wideband 
dOOp  in  2  da;.  .  27. 

12,  It'  5  BMO,  in  4  days  of  10  hours  <  I  bench  of  10 
Bee   of  hardness,  5<                             Wide,  and    (\\\\.    deep,    how 

many  mm  can  dig  a  trench  of  5  degree  of   hardn-  long, 

l.Ut.  wide. and  4 \ ft.  deep,  in  9  days  of  8^  hours  ea< 

I  pin  SO  in  1  year,  wliat  will  $900   train    in  8m.? 
11.   It'  $><><)  train    $1  I  riths,  what  will   $100  train  in  1 

1  .*».   If  $100  gain  $G  in  1  year,  in  what  time  will  $300  gain 

1G.  If  $100  gain  $G  in  1  year,  what  principal  will  gain  $12 
in  8  month-  ? 

17.  It'  a  2-penny  loaf  weighs  9oz.  when  wl  .  Gd.  per 
bii-hel.  how  much  bread    may  be    booghi   lor   3s  2d.  when  wheat 

L  per wmhcl ?  An*,  l  lib.  lOoz. 

18.  A  wall,  which  was  to  be  bnilt  .12  feet  high,  was  raited  I 
feet  by  B  m.n  in  12  dnjlj  how  many  men  must  be  employed  to 
build  the  remainder  of  the  wall  in  !•  <1  |  1. 

19.  If  f>bbl.  of  flour  serve  a  family  of  8  persons  12m.,  how 
many  bbl.  will  serve  a  family  of  12  persons  10  month-  ? 

.  If  16  horses  eat  21  bushels  of  oats  in  6  days,  how  many 
bnekeli  will  2->  boreea  eat  in  20  d 

21.  A  of  1600  men  have  bread  enough  to  allow  24 
ounces  per  day  to  each  man  for  3  hot,  the  garrison  being 

forced  by  400  men,  how  many  ounces  per  day  may 
man  have  in  order  that  they  mav  hold  out  against  the  enemy  30 
days? 

22.  If  3  compositors,  in  2  days  of  9  hours  -  type  for 
27  pagBtj  each  page  consisting  of  30  lines  of  45  letters  each, 
how  may  compositors  will  set  36  pages  of  40  lines  of  54  letters 
each,  in  6  davs  of  8  hours  each  ? 


COMPOIM'    PROPORTION.  267 

If  a  man,  walking   U  boon  I  day  for  8  days,  travel  384 
.  in  how  many  daya  of  10  boon  each  would  be  walk  240 
miles  traveling  at  the  same  rate? 

2  1.    It"  a   man   travel   880   miles   in  7  days  traveling  10  hours 
.   bo*   many  miles  will   he  go  in  12   days,  traveling  a* 
the  Sam<-  nie,  only  9  hours  eaeh  day  } 

It    12  horses  or  10  oxen  eat  2  tons  of  hay  in  8  wi 
how  mueh  hay  will  18  horses  and  2  eki  ? 

It*  it  take   88   reams  of  paper  to  make  1500  copies  of  o 
book  of  11   ■heeiBj  how  many  nam-  will  be  required  to   mako 
eopies  of  a  book  of  9  sheets? 

27.  It'  600  tile-,  each  12  inches  square  will  pave  a  court,  how 
many  tiles  that  are  1<»  inches  long  and  8  inches  wide  will  pave 
another  court  which  is  3  times  as  long  and  half  as  wide? 

28.  How  many  bricks,  each  8  inches  long,  4  inches  wide,  and 
2  inches  thick,  would  occupy  the  same  space  as  600  stones,  eaeh 
2  feet  long,  1^  feet  wide,  and  8  inches  thick  ? 

29.  If  7  shares  in  a  bank  yield  their  owner  $17.50  in  3 
months,  how  much  will  12  shares  yield  in  2  years? 

If  3  men,  in  16  days  of  12  hours  each,  build  a  wall  30ft. 
long,  8ft.  high,  and  8ft.  thick,  how  many  men  will  be  required  to 
build  a  wall  45ft.  long,  9ft.  high,  and  6ft.  thick,  in  24  days  of  9 
hours  each? 

31.  If  the  transportation  of  9hhd.  of  sugar,  each  weighing  12 
cwt.,  2<>  leagues,  cost  $50,  what  must  be  paid  for  the  transporta- 
tion of  50  tierces,  each  weighing  2^  cwt.,  300  miles? 

|800  gain  $18  in  9  months,  what  is  the  rate  per  cent.  ? 

33.  If  a  bar  of  silver  2ft.  lin.  long,  6in.  wide,  and  3in.  thick, 

26,  what  is  the  value  of  a  bar  of  gold  1ft.  9^,3 in. 

long,   sin.  wide,  and  4in.  thick,  the  specific  gravity  of  silver  to 

that  of  gold  being  as  10.47  to  19.26,  and  the  value  per  oz.  of 

silver  Wring  to  that  of  gold  as  2  to  33  ?  Ans.  $12> 

.;  1.   If  196  men,  in  5  days  of  12h.  6m.  eaeh,  dig  a  treneh  of  9 
:.et  long,  8]  feet  wide  and  4$  feet  deep, 
how  many  men  will  he  required  to  dig  a  trench  of  2  degfl 
hardness  168$   feet  long,  7$  feet  wide,  and  2J  tot  deep,  in  22 
days  of  9  hours  each  ?  Ans.  1 5. 


2G8  ALLIGATION. 


ALLIGATION. 

33*1.    Alligation  treats  of  mixing  nmple  -  of 

different1   qualities,  producing  a  compound  of  some  intermediate 

quality.     It  is  of  two  kinds,  Medial  and  Alternate. 

ALLIGATION  MEDIAL 

336.  Allh;  \  i  tOX  IfHXLiX  ||  tbi  proe. -<  liy  which  we 
find   the    price  of  the    mix:  R   the   quantities   and    pi 

of  the  nmfloi  are  giim 

!.  A  mi  reliant  IBtSM  5  gallons  of  oil  worth  4s.  per  gal. 
with  1  gal.  at  fo,  2  gal.  at  1  Is.,  and  .'5  gal.  at  12>.  What  is  the 
value  of  a  gallon  of  the  mixt 

5  gal.  at  At.  per  gal  are  worth  20s. 
4     u        5s.         «  "         20s. 

2     "      lis.         «  " 

_3     "      1-'.         u  "         36s. 

.-.  1  I  are  worth  98a. 

and  1  gat  M  worth  ^  of  98s=7-..  An-. 

All  example-  of  tin-  nature  are  solved  on  this  plan.     Hence, 

337.  To  find  the  price  of  a  mixture  when  the  num- 
ber of  articles  mixed  and  their  prices  are  given, 

Iiri.F..  Di ride  the  total  value  of  the  articles  mixed  by  the 
sum  nf  the  simples,  and  the  quotient  is  the  price  of  * 

_  A  miller  mixes  20  bushels  of  corn  worth  80c.  per  lnidi. 
with  1«) hush,  of  rye  at  $1,  40bnflh.  of  Oil  .  and  .''"bush. 

of  barley  at  0(>c. ;  what  is  the  price  per  bushel  of  the  mixture? 

3.  A  grocer  mixes  10  pounds  of  sugar  worth  6c. per  lb.  with 
1211).  at  8c,  41b.  at  12c,  and  51b.  at  15c. ;  what  is  a  pound  of 
the  mixture  worth? 

ALLIGATION  ALTERNATE. 

338.  Alligation  Altlrnatk  is    the    process   of  mixing 

335.    What  does  Alligation    treat  of!    Jt  is  of  how   many   kinds?     What? 
336.    What  is  Alligation  Medial?    33T.   Rule? 


AM. 
quantities   of  different    prices   so   M    to  obtain   ■   mixture   of  a 

loqiiircd  intermediate  prim.    Then  are  three  problems. 

3tli>.    Pboblhi  l.    The  prices  of  Beveral  kinds  of 

ascertain  how  much  of  each  kind 

may  •  impound  of  a  proposed  medium 

1.  A  farms*  wi-hes  to  mix  oats  worth  30e.  per  busk  with 
\\(,r:li  l  to  make  ■  mixture  worth  42a ;  how 

inaiiv  bushel-  of  each  may  In-   take? 

A\u.Mr  It  is  evident  that  be  must  mil  them  in  such  pro- 

tO  gain  just   M  much  M  his   oats   as   he   looei  00  the 

barley.  Now,  be  gains  12c  on  l  bush,  of  oats  and  totea  but  8c. 
oo  l  bash,  of  barley;  . .,  for  each  boahd  of  oats  he  most  take 
12  -r-  3=1  busheu  of  barley, 

■BOON   mi  i  hod. 

A9  (30-,       3  — 12c.  X     3= — 36c.,  deficiency. 

4*  |  .[:,  J     12  _|_  3c.  x   12  =  +36c,  surplus. 

Having  written  the  prices  of  the  oats  and  barley  in  a  vertical 
column  and  the  price  of  the  mixture  at  the  left,  as  above,  we 
write  the  difference  between  the  mean  price  (i.  e.  the  price  of 
the  mixture)  and  the  price  of  the  oats  against  the  price  of  the 
barley,  and  the  difference  between  the  mean  price  and  that  of 
the  barley  against  the  prke  Of  the  oats,  and  the  differences  stand- 
jain-t  the  prices  of  the  oats  and  barley,  rcspeetively,  will 
MO!  the  proportion!  ffUniHHm  of  Oatl  and  barley  to  be 
taten  1  for  it  will  be  seen  that  the  product  of  the  deficiency  in 
the  value  of  a  bothelof  oats  multiplied  by  the  number  of  bushels 
of  oats  ( —  12c.  X  3  =  — .">•><•.),  h  7y  equal  to  the  pro- 

dint  of  the  surplus  in  the  value  of  a  bushel  of  barley  multiplied 
by  the  number  of  bosheli  of  barley  (+3c.  X  12  =  -f-J 

Cat  two  products  are  composed  of  the  same  factors  ;  and 
one  representing  B  deficiency  and  the  other  a  surplus  they  will 
balance  each  other. 

In  tfai  ataas  manner,  any  number  of  pairs  of  simple-  in 


Wlmt  U  Alligation  Alternate?    IIow  many  Problems?     339.    01. 
Problem  1?    Explain  the  analysis  of  Ex.  1.    Explain  the  2d  method. 
23* 


270 


allr; 


made  to  balance,   as   in  Ex.  f,  the  price  of  one   simple   in  each 
pair  befog  tar  sad  thai  of  the  other  greater  than  the  mean  i 

In  performing  the  operation,  the  terms  are  O  bj  a  line 

merelj  for  convenience  of  reference  In  comparing  diem. 

2.  A  mereliant  has  4   kind  ST,  Worth    several!; 

.  and  16c.  per  lb. ;  how  mix  them  so  as  to  make  a 

mixture  worth  10c.  per  lb.  ? 

OPE  RATIO*. 


—30c 


6— .  C  —4c  X  6  =  —  24c 

8-,  8  —  2c.  X  3  =  -  6c.  , 

<  13J  2  +3cX2  =  +  6c)     ,  ,0 

1G-J  4  +6c  X  4  =  +24c  j  "T™0* 

Each  pair  of  these  products,  \  1  -t  and   4th,  and  the  2d 

and  3d,  will  necessarily  balance  ;  for  they  are  composed  of  the 
8am e  factors,  and  the  one  marked  -|-  represents  a  surplus  and 
the  one  marked  —  represents  a  deficiency.  By  this  method  the 
quantities  always  balai.  are,  however  many  simples  may 

il  in  the  mixl 
J!  10.    There  evidently  may  be  as  many  in<! 
all  correct,  a*  there  are  different  ways  of  pairing  the   nm] 
and,  by  taking  multiples  of  these,  the   results  may    b 
Indefinitely,  so  that  there  may  be  an  infinite  number  of  answers 
to  one  que  St 

Among  other  methods,  the  2d  example  may  have  the  foil 
solutions,  and  each  may  be  proved  correct  by  Alliyr 
ial 


10 


8-' 


1G— J 


311).  a] 

r  «=r 

-|      S+6=i91b.  :;■ 

61b."    8c 

10  . 

Gib."    8c 

41b.  "13c 

1  13-H-- 

J                     41b.  "13c 

lib,  •• 

[iS 

4  -f2  =  Gib.  "  1  :  . 

From  these  illustrations >: 

Rile.  Write  the  prices  of  the  several  simples  in  a  vertical 
column  ;  on  the  lefty  separated  by  a  line,  write  the  proposed 
medium  price  ;  connect,  by  a  line,  each  price  that  is  less  than  the 


339.    Explain  Ex.  2.    How  are  the  prices  connected?     How  do  they  balance? 
340.     How  many  answers  may  there  be?     How  proved  correct?    Eule? 


ALLIGATION.  271 

hi  with  <>ne  or  more  that  is  </rca!ri\  and  each  that  is  y 
one  or  more  that  is  faff  ,•  write  the  difference  f»rtic>  cu  the 
and  the  price  of  each  simple  against  the  >< 
or  nn  7,  which  the  simple  \  ■  I ;  these  differ* 

or  their  sum  if  two   or  more  stand  ai/ninst   one  price,  will  he  the 

urtionai  parts  of  the  several  simj>les  which  mat/  be  to 
form  the  jnixture. 

3-11.    Each  of  the  foregoing  method-  ii  simple  and  correct, 

but,  for  the  convenience  of  the  merchant,  there  is  I  better  mode, 
me  the  quantities  of  the  simples,  and  then,  by  calcula- 
tion, correct  the  assumption,  as  follow- : 

3.   A  merchant  has  0  kinds   of  wine  worth   .r>-.,  Bk„  8fc, 
and  1.3s.  per  pal.     "What  quantities  of  each  may  he  take  to  make 
a  mixture  worth  9s.  per  gallon  ? 

s.  pal.  s.  *. 

7  X  —  4sbs — 88 
G  6X—  3=—  IS       % 

8  6  X  —  1  =—  J>  —52,  deficiency. 

13  2X+4=+    8 

15  4  X  +  C  =  +  24  _j_32,  surpius. 

p;al.  s.         — 20,  deficiency. 

Add  wine  at  15s.,  4  X  +  6  =   -f-24,  surplus. 

-J-4,  surplus. 
Subtract  wine  at  13s., —  1  X  +  4=      — 4,  deficiency. 

Having  assumed  7gal.  at  5s.,  Ggal.  at  G^.,  figal.  at  8s.,  2gal.  at 
and  4gaL  at  1V,  we  find  the  mixture  is  not  worth  so  mueh 
as  it  should  be  by  20s.  Now  this  may  be  remedied  by  putting 
in  more  of  the  highel  priced  wines  or  tesa  of  the  cheaper.  If 
id  -Igal.  more  of  the  l5§,  wine,  this  will  balance  the  d<  li- 
cieiicy  and  OTIHtfl  a  surplus  of  Is.,  and  this  may  be  corrected  by 
taking  out  IgaL  of  the  13s.  -wine.  Thereare  now  in  the  mixture 
7gal  at  5s.,  6gal.  at  .  at  8s..  lgal.  at  13s.,  and  8gal.  at 

[5a, 

mark.     The  deficiencies   are   marked   by  the  >ign  —  and 
the  '  y  -|-  to  aid  the  mind  in  making  correctional 

341.    Another  method,  explain  it.    What  la  the  remark? 


9s. 


878  ALLIGATION'. 

Notk.  This  mode  of  correcting  may  be  indefinitely  raried,  hence  the 
merchant  may  take  the  simple*  in  a  ratio  more  nearly  as  be  desires  than  by 

fither  of  the  other  modes. 

Let  th«-  poj  -  Pirampl  b  of  die 

l    th.-m  ; 

4.  A  grocer  wishes  to  mix  teas  worth  2  .  and 

o  thai  tin-  compound  may  be  worth  pound, 

many  pounds  of  each  may  he  take? 

rth  $IG,  $20,  $28,  $  !<">.  and  $50  per 
1m  tad  ;  what  unmbei  of  each  may  he  sell  at  an  average  - 
$30  per  li 

:i  1*2.    Tuublem  2.    The  price  of  each  of  the  bud.] 

the  price  of  the  compound,  an<i  the  quantity  of  one  1. 
being  given,  to  find  how  much  of  each  i  nm- 

jiles  may  be 

Rule.  Find  the  proportional  parU  as  in  the  preceding  Prob- 
lem ;  then  say,  at  the  proportional  part  of  thai  simple  whose 
quantity  is  given  is  to  the  given  quantity,  so  is  each  of  the 
other  proportional  parts  to  tlie  required  quantity  of  each  of  the 
otlter  simples,  severally. 

Ex.  1.     How  many  pounds  of  sugar  at  4,  6,  9,  ami  10c.  per 
lb.  may  be  mixed  with  121b.  at  13c,  to  make  a  compound  \\ 
8c.  per  lb.  ?  Ans.  15,  9,  6,  and  6  i  ind  10c 

c-  lb.  if  we  connect  the  pricei  at  m 

2  +  1=3         and    411>.    for    tin      g 

parts.     Now  if  the  41b.  at   I$c 
2 

*         will  become  12  11-..  (/<>■  given  owftv 
and    if    each    bf    tlit- 
proportional  parts  be  increased  in  the  same  ratio,  evidently  the 
per  lb.  or  the  mixture  will  remain  unaltered  ;  thus, 

41b.  at  ISe.  :  121b.  at  18c,  :  :   51b.  at  4c.  :  151b.  a 
41b.  at  1.)  •   :  121b.  at  13c  :  :  31b.  at  6c  :    91b.  at  Cc. 
etc. 

341.  What  is  the  Note?    34  i.  Object  of  Problem  2?    Bttlef    I.xj.lanatiou? 


8c. 


L0- 


AU.I';\T!"\. 

2.  How  many  gallons  of  win.*  at  %  10,  and  15s.  per  gal.  may 
he  mixed  with  L5gaL  of  water  of  no  exchangeable  vain.-,  to  make 
a  mixture  worth  1--.  par  gal.? 

How   many  11).  of  wool   at   10,   10,  an<l   56c  per  H).  mav  be 

1  with  2  111*,  at  -loo.  to  make,  a  mixture  worth   llV.  per  lb.? 

3-13.    Problem  8.    The  prices  of  the  several  simple*, 

price  of  the  compound,  and  the  entire  quantity  in  the 

ling  given,  to  Had.  how  much  of  each  simple 

i6  taken  : 

II:  !  i  .       /' '.■/./  the  proportional  parts  as  in   Problem   1  ;  then 

say,  as  the  gum  of  t/te  proportional  parts  is  to  eompotmdf 

rack  of  Hie  proportional  parts  to  the  required  quantity  of 

each. 

1.1  have  4  kinds  of  coffee,  worth  8,  11,  14,  and  20c.  per 
pound  ;  how  many  pounds  of  each  may  I  take  to  form  a  com- 
pound of  Go  lb.  at  18a  per  lb.? 

Ans.  28,  4,  8,  and  201b.  at  8,  11,  14,  and  20e. 

c.  lb. 


13c. 


8—, 

ft 

20- 


i:>  :  CO  lb.  ::  71b.  :281b.  at  8c 
15  :  601b.  ::  lib.  :    41b.  at  lie. 
etc 

TVe  find  that  the  sum  of  the  proportional  parts,  if  linked  as 

.is    15  lb.,  and   if  this   be   quadrupled,   GOjb  ,   the   required 

comjKjund,  will  be  obtained  ;    but  the  whole  compound  will  be 

quadrupled  by  i  each  of  the  proportional  parts  in  a  four 

fold  ratio. 

How  many  ounces  of  gold,  that  if  1G,  18,  20.  and  21  carat* 
nay  be  taken  to  form  a  i  .'1  raruts  fine? 

3.   How  many  -hep  worth  9,  12,  16,  18,  and  2  U  each,  may 
be  taken  to  form  a  Hock  of  125  sheep  worth  15s.  each? 

343.    Object  of  rrob1«m  8?    Kulc!    Explanation  ? 


271  INVOLUTION. 


INVOLUTION. 

344.  A  Power  of  a  numl.. ■  ||  ih--  product  obtained  by 
u-iiiLT  the  number  two  or  more  timet  as  a  factor. 

Involution  is  the  process  of  raising  a  number  to  a  p< 

The  numlx  r  involved  is  the  1*/  power  of  it.-rlf.  It  is  al.-o 
the  root  of  the  other  powers  (Art.  1 1  J  ■'>  and  C). 

:M»1    The  Index  or  Exj  t  ■  power  is  a  figure 

placed  at  the  right  ind  ■  little  above  the  root  to  show  how  many 
times  it  is  used  as  a  factor  (Art.  112,  Note  1)  ;  thus, 

4X4=      16  =  4,,i.e.  tli<- l>,1  ji(,w,Tor^(juareof4. 
4x4X4=     64=4*  i.<-.  the  M  power  or  cube  of  i. 
4x  4X  *X  4=  256=  t4.  Le,the  4th  power  of  I. 
4X4X4X4X  4  =  1024  =  4*,  i.e.  the  .r,th  power  of  4 

1 1    nee, 

IM6.     To  involve  a  number  to  any  requited  power, 
Rule  1.      Write  the  index  of  the  potver  or, ,-  (kt  root ;  or, 
Rule    2.     Multiply  the  number  by  itself   ami   (if  a   //if/her 
power  than  the  second  is  required)  multiply  this  product  by  the 
original  number,  and  so  on  until  the  root  has  b<  as  a  fac- 

tor as  many  times  as  there  are  units  in  the  index  of  the  required 
power. 

Ex.  1.  What  is  the  3d  power  of  6  ? 

Ans.  C3=6  X  6  X  6  =  21G. 

2.  What  i>  the  5th  power  of  3  ?  Ans.  243. 

3.  What  is  the  4th  power  of  5  ? 

4.  What  is  the  8th  power  of  2  ? 

5.  What  is  the  2d  power  of  10  ?  Ans.  25  G. 

6.  What  is  the  3  power  of  J  ? 

4__2     f2]s 2       2       2_28___8_ 

6""3;  UJ  —  3"  X  3  X  3  ~33  —  27* 

344.  What  is  a  Tower  of  a  number?  What  is  Involution?  Wliat  is  the  num- 
ber involved?  343.  What  is  the  Iudcx  or  Exponent  of  a  power?  345.  Rule 
for  involution?     Second  rule? 


iNvm.rr 

7.  What  h  the  M  power  of  ^  ? 

8.  What  k  the  N  i-ow.r  of  J  ."< 1'7. 

*j.  What  h  th<-  hh  power  of  .12  ? 

10.  What  is  the  tqpart  of  I 

H»iJ  and(S)«  =  U  =  5A.  Ans. 

11.  What  is  UM  .Mil...  of  3fc? 

l.    It  will  be  observed  that  ■  nixed  Mother  is  fh-t  redtteed  to  an 
■■proper.  ftfccoToo,  ami  ■  eoonooo  titanium  m  reduced  to  its  lowest  tcnns, 

uml  then  each  term  is  involved  separately.     AleO  that  the  number  of  tleci- 

nial  j.laees   in  the   power  of  a  ileciinal   is  eipial   to  the   nuinher  of  decimal 

In  the  mot  multiplied   bj  the  index  of  the  pOWOff  (Art.  171). 

N-ii:  2.     The  powers  of  a  number  greater  than  unity,  are  greater  than 

>t,  and  the  powers  of  a  pro|.er  fraction  are  less  than  the  root  ;   thus, 

the  cube  of  2  is  8,  which  is  greater  than  2 ;  and  the  square  of  i   is  J,  which 

is  greater  than  3  ;  but  the  square  of  Z  is  4,  which  is  loM  than  3. 

#17.    To  multiply  different  powers  of  ili«-  mme  num- 
ber  together : 

II f  i .i-:.     Add  the  i/i<lices  of  the  factors  together,  and  the  sum 
will  be  the  index  of  the  product. 

12.  Multiply  the  square  of  3  by  the  cube  of 

32x3,  =  35;i.e.  3X  3  X3X3  X  3  =  35=  2-13. 

13.  Multiply  52  by  54.  An- 

JM8.     To  involve  a  quantity  that  is  already  a  power  : 

Kri.i:.     Multiply  the  index  of  tli>  lumber  by  the  index 

of  the  power  to  which  it  is  to  be  rot 

Tims,  the  3d  power  of  2*  is  26,  for  22  =  2  X  2,  and  the  3d 

power  of  2  X  2  is  2X2  X  2x2  X2x2  =  2x2x2X2X 
2  X  l>  =  26=64. 

1  J.  What  is  the  1th  power  of  3s?  Ans.  312. 

15.   What  is  iheJUh  DO* 

340.    What  is  done  with  a  mixed  number?    How  is  a  common  fraction  in- 
•v  many  decimal   places  in  the  power  of  a  decimal I     If  the  root  is 
greater  than  one,  are  its  powers  greater  or  less  than   the   i » •« «t  *     If  tlie  root  is 
less  than  one?    317.    Itule  for  multiplying  different  powers  of  the  same  num- 
ber together?    348.    Rule  for  involving  a  power? 


276 

!M9.     To  divide  a  power  of  a  number  by  any  other 

r  of  the  same  numl 
BULK.      Subtract  the  exponent  of  the  divisor  from  the  exponent 

of  the  dividend. 

1 G.  Divide  5r  by  5s.     Ans.  5T  +  58  =  5«,  for  57  -f-  5s  =  -  = 

5X5X5g5X5X5><5  =  5X5X5X5  =  5-6-" 

1 7.  Divide  8*  by  8r.  18.  Divide  4T  by  4s. 


B VOLUTION 


3«?0.  A  root  of  a  number  is  one  of  the  equal  factors  whose 
continued  product  is  that  number  (Art.  1  '■)). 

Involution   or  Extracting    EtoOTf  is  the  resolving  of  a 
quantity  into  as  many  equal  factors  as  Oiere  are  units  in 
of  the  root. 

tttil.  There  are  two  method-  of  indicating  a  root,  one  by 
BMiM  of  the  radical  sign,  ^,  and  the  Other  by  means  of  a 
tonal  in--' 

The  figure  placed  over  the  radical  sign  is  the  index  of  the 
root,  and  is  always  the  same  as  i  irfimtftff  of  frmtJ 

index  ;  thai,  the  cube  root  of  8  is  \/8  or  8.* 

Tlie  square  root  of  the  cube  of  4,  or  the  cube  of  the  square  root 
of  4,  is  v/43  or  4*. 

If  no  number  is  over  the  radical  sign,  2  is  understood. 

35*2.     Evolution  m  the  reverse  of  INVOLUTION. 

In  Involution  the  root  is  given  and  the  power  required. 
In  Evolution  the- power  is  given  and  the  root  required. 

349.  Rule  for  dividing  one  power  by  another  power  of  the  same  number! 
350.  What  is  a  Root  of  a  number?  What  i*  Evolution?  351.  How  many 
methods  of  indicating  a  root?  What?  What  is  the  index  of  the  root?  What 
of  the  index  2? 


UOOT. 

3*13.      All  numbers  can  be  "!/  required   power, 

but  comparatively  fnc  can  1>< 

Thotc  Quintan  which  can  li.-ivr  th.ir  roots  extracted  are  called 
perfect  powers,  ami  their  root-  are  rational  Bombers.     Nun 

,:mot    ta    I  called    imperfect  power.-,  and 

their  roots  arc  irrational  or  turd  osnabt 

A  Dumber  may  be  a  perfect  power  of  one  naflbe  ot  d 

ami  an  imperfect  power  ot*  another;  thus.  16  is  a  perfect  aqoare, 
hut  an  imperfect  cuhe.  whereas    i_>7  ifl  I  ita,  hnt  an   Km* 

|H|    6  1    ifl   a   p  re,   cuhe,   and 

power. 

3*7-1.  Kverv  power  and  every  root  of  1  U  1.  There  is  no 
other  number  whose  powers  and  roots  are  all  alike. 

The  roots  of  a  proper  fraction  are  greater  than  the  fraction, 
and  the  roots  of  any  number  greater  than  unity  arc  leal  than  the 
number  ;  thus,  ^/J  =  Jj,  which  is  greater  than  |;  \/{jJ=j», 
which  exceeds  £J  ;  but  */%$  =  £,  which  is  leu  than  y  ;  \/.S 
C3  2,  which  ifl  toflfl  than  8.    . 

EXTRACTION  OF  THE  SQUARE  ROOT. 

3*7*7.  To  kxtkact  tiik  SQUARE  ROOT  of  a  number  is  to 
resolrc  it  mio  Iwo  equal  farfnrs.  i.  e.  to  find  a  number  which, 
multiplied  into  itsrlf  will  produce  the  given  num: 

3*70.     The  tqmart  of  a  number  consists  of  hcice  as  many 
figures  as  the  root,  or  of  one  less  than  twice  as  many  ;  thus, 
Ro-  1,        9,         10,  99,         100,  999. 

1,      81,       100,      9801,    10000,    998001. 

Hence,  to  ascertain  the  number  of  figures  in  the 
square  root  of  a  given  number, 

353.    Can   til   number*  be  involved'     Evolved?    Wbat  are  perfect  powers! 
Rational  ro<  '  feet  powers?    Irrational  or  surd   roots?    May  a  i 

be  a  perfect  power  of  one  degree  and  an  imperfect  power  of  another  degree? 
A  perfect  power  of  several  degrees?  Illustrate.  15*.  What  of  11  The  roots 
of  a  proper  fraction,  are  they  greater  or  less  than  the  fraction?  The  roots  of  a 
number  greater  than  one?  355.  To  extract  the  square  root  of  a  number,  what? 
856.    How  many  figures  in  the  square  of  a  number? 

24 


278  EVOLITI 

Rile.     Beginning  at  the  right,  point  off  the    number 
periods  of  two  figures  each,  and  there  trill  be  one  figure  in  the 
root  for  each  period  of  two  figures  in  Ute  square,  and  if  there  is 
an  odd  figure  in  Ute  square  there  will  be  a  figure  in  the  root  for 

that. 

Ex.  1.  How  large  a  square  floor  can  be  laid  frftft  iare 

feet  of  boa: 

If  we  knew  the  length  and  breadth  of  a  floor,  we  should  find 
it^  area  by  multiplying  tin-  length  bj  the  breadth  (Art  KM). 
<>r,  in  this  example  (since  length  and  breadth  an*  equal),  by 
multiplying  the  length  by  itself.  But  tee  are  now  to  reverse  this 
process,  and,  knowing  the  area,  to  fin  A  ,'h  of  one  side. 

Since  the  dumber,  .">7G,  consists  of  thro  ;<K.t  will 

consi  ires,  tens  and  units,  and  the  square  of  the 

must  be  found  in  the  5  (hundred-). 

OPERATION. 

•  Now  the  square  of  2  (:  I  (hun- 

*>'6(24  U)  and  the  square  of 

_ (hundreds) :  and,  a-s  5  (hundred^)  i^  i«->s 

|  1)17  6  than   lJ   (liu:  can  be  but   I 

17  6  -)  in  the  root.    Let  na  now  construct 

7  a  square.  Fjg,  1.  •  of  which  shall 

be  S  )  jn  length. 

area  of  this  square  is  20X20  =  400  square  feet,  which,  dedu 
from  .  will  leave  176  square  I  seed  in  enlarf 

floor.  To  preserv.-  the  square 
form.  tln>  addition  most  be  i 
uj>on  the  4  sides  of  the  floor,  or, 
qnmlly  uj>on  2 
ea,  a^  in  1 
the  nature  of  the  oaCBj  the  %  addi- 
tions, bin 

breadth  ;  and,    if    th«'ir   length 
known,    we    could    determine    tin  ir 
idth  by  dii  iding  their  area,  I  ~  6 
feet,  (An.    102). 

But   we  do  know  the  length  of  bh  -\-  cr,  viz.  twice  the  tens  of 

350.    To    a.-certaiu   tue  cumber  of  figure*  in  a  square  root,  Rule?    fcxpl&in 
Ex.  1. 


r 

Fig.  2. 

n 

m 

90 

4 

4 

4 

80 

1  6 

d 

k 

c 

_i 

«^ 

20 

20 

c 

4 

80 

20  1 

>     4  ft 

'J 

the  root  =  4  (tens  or  40  ft),  and  this   h  suflcleutlj  near  to  the 

whole    leagtfc   of  the    addition- 

tO    Serve     M    I     trial     d 

Nuw     17»;  ~  40,    or,    what     is 

.one    in     effect,    17-^-4, 

l   feel   fbr  the  breadth 

of  tin-  addition,  and  this  Added 

tO     the     trial     dieisor,     10,    or 

annexed  to  tin-    l   (tent)  will 
1 1,  the  whole  length  of 

bm  -f-  CT,  the  //  .    And 

41  x  4=  L76]  i.  e.  the  length 

of  the    addition    multiplied    by 

it-  braadtfa  givei  it-  .■ 
It  will  be  seen  thai   eyery 
foot  of  hoard  is  used,  and  the  floor  is  a  square,  each  side  of  which 

is  20  +  4  =  24ft.  long,  Ans. 

*,\r>7 .  The  same  species  of  reasoning  applies,  however  many 
figures  there  may  he  in  the  root.     Hence, 

To  Extract  the  Square  Root  of  a  number, 

Kile.  1.  Separate  the  given  number  into  periods  of  two 
figures  each,  by  placing  a  dot  over  units,  h/utdreds. 

2.  Find  the  greatest  square  in  the  left-hand  period  and  set  its 
root  at  the  right,  in  the  place  of  a  quotient  in  long  division. 

3.  Subtract  the  square  of  this  root  figure  from  the  lrj't-ha/id 
period,  and  to  the  remainder  annex  the  next  period  for  a  divi- 

d(/td.  , 

4.  Double  the  root  already  found  for  a  TRIAL  divisor,  ami, 
omitting  the  right-hand  figure  of  the  dividend,  divide  and  set  the 

II  the  next  figure  of  the  root.     Also  set  it  at  the  right 
of  the  trial  divisor,  and  so  form  the  tiui:   divisor. 

5.  Multiply  the,  true  divisor  by  this  new  root  figure,  arid  subtract 
the  product  from  the  dividrnd. 

6.  To  the  remainder  annex  the  next  period  fi>r  a  new  dividend, 
double  the  part  of  the  root  already  found  for  a  trial  divisor,  and 
proceed  as  before  until  all  the  periods  have  been  employed* 


3.7.    Kule  for  extracting  the  square  root  of  a  number? 


230 

Note  1 .     The  left-hand  period  may  consist  of  bat  one  figure. 

Note  2.  The  trial  divisor  being  smaller  than  the  true  divisor,  the  quo- 
tient is  frequently  too  large,  and  a  smaller  number  must  be  set  in  the  root.  This 
usually  occurs  when  the  addition  to  the  square,  a  e,  is  wide,  and,  conse- 
quently, the  square,  h  n,  large  ;  or,  in  other  words,  when  the  trial  divisor  is 
much  less  than  the  true  divisor. 

358.    Proof.    Square  the  root;  thus,  in  Ex.  1,  24s  s= 

2.  What  is  the  square  root  of  401956? 

OPERATION. 

4  6  1  9  5  6(6  3  4,  Ans. 
86 

12  3)419 
3  6 


1  2  6  4)  5  0  5  6 
5<' 


3.  What  is  the  square  root  of  1918-1 1  ?  Ans.  438. 

4.  What  b  the  square  root  of  677.4i- 

5.  What  is  the  square  root  of  67081  ? 

OPERATION. 

B  9,  Ans.  In  this  example,  the 

4  hand  period  consists  of  but 

4  5)2To  one   fi£ure«      So»   also,   the 

2  25  tr'a*  divisor,  4,  is  ftflfltfriH 

.  tim**-  :  and  the  2d 

5  0  9)4581  remainder,   4.">,  equals    fie 

.  the  true  root 
0  re  is  but 

&   What  is  the  square  root  of  97  Ans.  31 

7.   What  is  the  square  root  of  136161  ? 
&   What  hi  the  square  root  of  4201C 
I.    W "hat  is  the  square  root  of  4304C7 

10.  What  U  the  square  root  of  22014> 

11.  What  is  the  square  root  of  1522756  ? 

12.  What  is  the  square  root  of  18671041  ? 

13.  What  is  the  square  root  of  60910: 

357.    What  is  Note  1*    Note  2!     358.    Proof?    Explain  Ex.  6* 


SQUARE  ROOT.  281 

l  i.  What  is  the  iqutre  root  of  16777116! 

Ol-I 

167  77216(4096,  figure 

1  6  '.  m  in  tin-  example, 

809)7772  ""  x  °  lu 

-  a  a  j  AM    trial     divi-or,    ; 1 1 1 . 1 

bring  down    tin-   next 

8186)49116  period  i<>  complete 

49116  dow  dividend. 

0 

15.  What  is  the  square  root  of  5701801  ? 

16.  What  is  the  square  root  of  1048.*>7 

17.  What  is  lie-  square  root  of  282  [TSt  19  ? 

Note  3.     Iii  tlfmiliifl  tlio  root  of  a  dicimnl,  put  tin-  first  point  over  hurt- 
iredths  and  point  toward  tin-  rigkt,  ami  if  tin-  bM  ptriod  is  nut  full,  ■aatt  0. 

18.  What  is  the  square  root  of  .4096?  .64. 

19.  What  is  tlie  square  root  of  .«'• 

20.  What  is  the  square  root  of  39.0Cl'  An-.  .;.•_»:.. 

21.  What  is  the  square  root  of  6046.6176? 

&  What  is  the  iqHan  root  of  f>.  Ans.  2.36+. 

OPERATION. 

SO  (  2.3  6  -j-  If  there  is  a  remainder  after 

4  employing   all    the    periods    in 

4  3)160  t'l("  "'m'n  <x:linI>^''  ,nc  opera- 

'  ,  ..  tion  may  be  continued  at   plea- 

sure   by    annexing     Miccc-ivr 
466)3100  j„  Hods  q£  ciphers,  decimally ; 

2  7  9  6  there    will,    however,    in    such 

472    )3040  0.  IT— pUie.  cttwam  be  a  renuiin- 

r  the  right-hand  ligure 

of  the  dividend  is  a  cipher,  whereas  the  right*hand  figure  of*  the 

subtrahend  is,  escesiarifo  the  right-hand  figure  of*  the  square  of 

some  one  of  the  nine  significant  figures,  the  right-hand  figure 
of  the  reel  "/"/  of  the  divisor  being  ahocyt  aKhe.  Now,  no  one 
of  these  nine  I  I,  will  give  a  number  ending  with 

a  cipher;  .-.,  the  last  figure  of  the  dividend  and  of  the  subtra- 
hend being  unlike,  there  wmti  oe  a  rtonainder. 

What  is  tb  An-.    1.411-J1+. 

358.    Explain    I  a    1   -We  3?     Explain  Ex.  22. 


282 


EVOLl 


I  L  What  fa  tli 8  square  root  of  20?  .  172-f-. 

What  is  the  square  root  of  31 G? 
20.  What  II  thr  .Mjuare  root  of  31. 

3*19.    To  extract  the  root  of  a  common  fraction,  or 
of  a  mixed  number: 

Kile.     Reduce  the  fraction  or  mixed  number  to  its  simplest 
form,  and  then  t<i!;c  the  root  of  the  numerator  and  denominator 
separably  ;  ur,   if  rm  of  the   fraction,  when  reduced,  it 

an  imperfect  square,  reduce  the  fraction  to  a  decimal  (Art.  \ 
and  then  proceed  as  in  the  foregoing  examples. 

27.  What  is  the  square  root  of  f  J  ? 

lie  square  root  of  {  Ans.  f. 

tbfl  square  root  of  j$r 
80.  What  is  the  square  root  o 

31.  What  is  the  square  root  of  10&  ? 

jiiare  root  of  §  of  $f? 
83.  What  is  the  square  root  of  $  ?  Ans.  .6o4-{-. 

23 

34.  What  is  the  square  root  of  =±  ? 

Application  of  the  S«x  :  >ot. 


360.    ATin  am;  l  Lis  a  figure  bounded 
bj  tfiOM  straight  linjBS. 

A  right-angled  triangle  has  one  of  its 

-  a  right  angle,  as  at  C 
The  side  op  the    rijrht   angle    is 

1  (be  hypotkmum ;  tlie  other  two 
are  the  base  and  perpendicular. 


li 


Bom. 


359.    Rule  for  extracting  the  root  of  a  common  fraction  or  mixed  number? 
*GO.    What  is  a  Triangle?     A  right-angled   triangle?     Hypothenuae?    Base? 


288 


u 

The  square  described 
on  the  hypothecate  of  a 
right-angled  triangle  H 

im  of  the 

ret  described  oo  the 

:  two  sides.  Also 
the  square  of  either  of 
the  t\  hich  form 

the  right  angle  Is  equal 
to  the  square  of  tin-  hv- 
pothcnosc  diminished  by 

qnarc  of  the  other 
tide.  Thi>  will  be  seen 
l>y  pointing  the  small 
squares  in  the  square  of 
the  hypothenuse  and 
those  in  the  squares  of  the  other  two  sides.     Hence, 

1st.  To  find  the  hypothenuse  when  the  base  and  per- 
pendicular are  given, 

lit  i.k.  Add  the  square  of  the  base  to  the  square  of  the  per- 
'pendicular,  and  extract  the  square  root  of  the  sum. 

2d.  To  find  either  side  about  the  right  angle  when 
the  hypothenuse  and  the  other  side  are  given, 

BULB.  From  the  square  of  the  hypothenuse,  subtract  the 
stjuctre  of  the  other  given  side,  and  extract  the  square  root  of  the 
remainder. 

.  l.    The  base  of  a  right-angled  triangle  is  6  feet  and  the 
dicalar  U  8  feel :  what  ii  (be  hypothem 

(V-'  =  ;io\  8*3x64*  36  +  6-4  =  100;  */100=10.     Ans.  10  ft. 
2.  Tin*  hypothennse  of  a  right-angled  triangle  ia  15  and  the 

H   12:   what  i<  the  perpendicular  ? 

=  225,  12»=  ill;  225—  111=81;  a/81  =  9,  km. 

300.    The  square  of  the  hypothenuse  equals  what?    The  square  of  out 
Other  -  may  this  appear?    Rule  for  finding  the    hypothenuse?    Base 

*r  Perpendicular?     Explain  Ex.  1. 


284  utimw 

3.  A  ladder  resting  upon  the  ground  21    fed   from  a  he.- 

just  reaehei  a  window  which  k  18  feel  hflghj  li<>w  bog  ii  the 

ladder? 

4.  A  tree  that  was  G4  feet  liLdi  U  broken  off  I  1  feel  high, 
part  broken  oil'  turning  iij><>n  the  still)  as  upon  a  hinge;  at  what 
distance  from  the  bottom  of  the  tree  doe*  the  lop  rtrike  1 1 1 . - 
ground  ? 

5.  Two  vessels  nail  from  tin-  same  pott,  DM  due  000l  40  ■ 
and  the  Other  due  -<>utli  9  mfUnj   be*  tar  apart  an-  they? 

6.  A  general  has  9801  men  ;  if  he  places  them  in  a  square, 
bOW  many  will  there  bi  bO  rank  and  file? 

7.  HOW  many  rod>  of  liner  will  1,<  n-«|uir«  d  to  indoor  CIO 
acres  of  land  in  a  square  form  } 

8.  A  farmer  sets  out  an  orchard  of  GOO  trees  so  that  the  num- 
ber of  rows  is  to  the  number  of  trees  in  a  row  as  2  to  3. 
trees  are  apart  and  no  tree  is  within   12\  feet  of  the 
fence;  how  many  square  feet  of  land  in  tl 

Fig.  3.  IUM.     In    li  ihined 

ar/  .  1  •»'.»),  a  sqwm    (Art.  1<»1 . 

.nil  right  mi  fled  tri Hu- 
ghs.     Tie-   line    AC   Kl    tin  .•    of 

the  square  and 

tl>.  /it/pothenuse  of  each  of  the  triang 

The  square  is  said  to  bo  mierib 

circle    and    the    i 

about  the  sop; 
The  diameter  of  any  circle  i-  to  it-  circumference  in  the  r 
of   1    to    &.14J  .rly  |     benet    tlie    diameter    multiplied   }>y 

"lll-V.ej    >vill    giro    the    circumference,  and   the    circumference 
divided  bj  3.141592  will  gi\.-   T 1  * « -   diane 

The  area  of  a  circle  may  be  found  by  multiplying  the  square 
of  it<  diameter  by  .785898,  nearly,  and  if  the  area  i>  divided  by 
.785398,  the  quotient  will  be  tlie  Bquare  of  the  diameter. 

301.  What  does  Fig.  3  represent?  What  is  the  line  AC!  What  is  said  of 
the  square?  Of  the  circle?  Ratio  of  diameter  to  circumferenc.  >  H..w  i<  cir- 
cumference found  when  diameter  is  given?  Diameter  when  circumference  is 
given?    Area  of  a  circle,  how  found?    Diameter,  when  area  is  given? 


SQUARE   ROOT.  285 

3<>£.     Similar  figures  are  figures  that  are  of  pCfinJOOlj  flip 

form,  whether  large  or  small. 
The  areas  of  all  similar  figures  arc  to  etch  otlicr  as  the  squares 
of  their  corresponding  li: 

What  is  tBO  diameter  of  I  circular  pood  which  -hall  contain 
HOB  M  much  area  as  one  8  rods  in  diameter  \     An-.  lOrd 

10.  The  area  of  a  triangle  H  '1\  square  inches  and  one  si<lc  of 

9  inches;  what  u  the  eorrespooding  tide  of  a  similar  tri- 
angle containing  '.»<;  square  inehe-  \ 

11.  What  is  the  side  of  a  square  that  -hall  contain  86  times 
a-  much  area  as  one  who-e  -ide  is  5  fee 

12.  What  i-  the  side  of  a  square  equal  in  area  to  a  circle  100 

u  diameter?  Ana.  88.622sq.ft. 

1$.  A  circular  field  contains  10  acres;  what  is  the  length  of 
it<  diameter? 

11.  What  i>  the  difference  in  the  expense  of  fencing  a  circu- 
lar 10-acre  lot  and  one  of  the  same  area  in  a  square  form,  the 
fence  co-ting  7ac.  per  rod? 

1  ">.  If  a  pipe  3  inches  in  diameter  will  empty  a  cistern  in  8 
minutes,  what  is  the  diameter  of  the  pipe  which  will  empty  it  in 
18  minute 

16.  The  area  of  a  rectangular  piece  of  land  (Art.  101,  Note) 
is  50  acres,  and  the  length  of  the  piece  is  to  its  breadth  as  5  to  1 ; 
what  are  the  length  and  breadth? 

17.  A  room  is  16ft.  long,  12ft.  wide,  and  9ft.  high;  what  is 
the  distance  from  one  lower  corner  to  the  opposite  upper  cor- 

Ans.  21.931+ft. 

18.  The  diameter  of  a  circle  is  10  inches;  how  many  inches 
in  the  side  of  the  inscribed  square?       Ans.  \/')0  =:  7.07 1-[-. 

By  figure   3   it   is  seen  that  the  diameter  of  the 
circle  i>  the  hvpotheiiu-e  of  a   right-angled   triangle  \vho<e  otlicr 
•  qual  to  each  other ;  •••  the  stpiare  "f  mtktr  side  of  the 
■If  <f  the  square  of  the  diameter. 

19.  What  b  the  side  of  the  greatest  square  >ti<-k  of  timber 
that  can  !>•■  hewn  from  a  log  18  inches  in  diameter? 

369.  What  are  rimilar  figures f    The  ratio  of  the  areas  of  similar  figures! 


J- 1 

EXTRACTION  OF  THE  (  DBE  BOOT. 

363.  To  extract  the  Cube  Root  of  a  number  is  to 
resolve  it  into  3  equal  factors  ;  t.  e.  to  find  a  numher  w/tirh,  mul- 
tiplied into  its  square,  will  produce  the  gi<  :>er. 

364.  The  cube  of  a  number  consists  of  three  times  as  many 
♦'S  as  the  root,  or  of  one  or  two  less  than  ti.  «  as 

many  ;   ti. 

B^ota,      1.       '\  10,  100, 

(ul....     1.     729,       1000,      97  1000000,     99700l' 

Hence,  to  ascertain  the  ires  in  the  cube 

root  of  a  L'iwn  Hum 

/  at  the  right,  point  off  the    numh,,-   into 
periods  of  three  figures  ea>-  rein  the 

root  for  each  period  of  three  figures  in  fkk 

one  or  two  figures  besides  full  periods  in  the  cube,  there  will  be  a 
figure  in  the  root  for  this  part  of  a  period. 

1.  Suppose  we  have  74088  blocks  of  wood,  each  a  cubic 
inch  in  Mze  and  form,  how  Ufrgt  a  cubical  pile  can  be  formed  Un- 
packing these  blocks  tog 

OPERATION. 


Trial  divisor.    1  8  0  0  , 
240 
4 


7  4  0  8  8(42  Root. 
64 

10  0  8  8  Dividend. 
10088 


True  divisor,  5  0  4  4  J  0 

As  there  are  tiro  periods,  the  root  must  consi-t  of  two  figures, 
tens  and  units  ;  and  we  seek  the  cube  of  the  tens  in  the  left-hand 
>  1  ;  the    gj  ube   in    74   is  M  root   is  4.      We 

the  root,  I.  at  the  rijrht  of  the  number,  and,  having  sub- 
tracted th  I, from  the  left-hand  period,  we  annex  the  next 
period  to  the  remainder,  10,  making  10088  for  a  dividend. 

303.  To  extract  the  cube  root  of  a  number,  what?  364.  How  many  figures 
in  the  cube  of  a  number?  To  ascertain  the  number  of  figures  in  a  cube  root, 
Bute?     Explain  Ex.  1. 


CUBE   RO«»T. 


Ffe  1. 


40  inches. 


Thus,  by  Ming  0  1000  of  the  blocks,  a  cube  is  formal  (Fig.  1  ) 

wfcoaeedge  ii  i<>  inches  andwhcee 
■jopteti  arc  6 1000  solid  be 

and   there   are  10088  blocks  re- 
OMiniog,  with  which  to  anlai 
ibio  pile  already  EbrnefL 
In  enlarging  this  pile  and  i 

tng  the  cubic  form,  the  nddi- 

must  be  made  upon  each  of 

the    8    laces,    or,    more    coim-ni- 

ently,  equally  opoa  any  3  adja- 
cent d,  />,  and  r,  as   in 
2.     What  may  be  the  thick- 

of  the  addition?      By  divid- 

iog  the  contents  of  ■  rectangular 

solid  by  the  area  of  one  ftee,  we 
obtain  the  thickness  (Art.  105)  ;  now,  the  remaining  10088  soli4 
inches  are  the  contents,  and  the  sum  of  the  area-  ot  the  .'5  square 
faces,  a,  b,  and  c,  is  sufficiently  near  the  area  to  be  covered  by 

the   additions  to  form  a 
*"g«   ••  trial    divisor;     for    the 

40  3  additions,  a,  6,  and    c 

(Fig.  2),  are  the  same 
as  one  solid  40  inches 
wide,  3  times  40  inches 
long,  and  of  the  thick- 
ness determined  by  trial. 
The  area  of  these  3 
is  the  square  of 
4  (tens),  which  is  16 
(hundreds),  multiplied 
by  3,  which  gives  4800; 
i.  e.,  to  obtain  a  trial  di- 
visor, we  square  the  root 
BgOffe  and  annex  00 
cau-e  the  root  figu; 
ten )  for  the  area  of  one 
and    then    multiply   this    area   by  .'!.      Dividing    10088   by 

obtain  the  quotient  2,  fur  the  ikicknm  of  the  tuUdi 
■  v  the  imit  figure  ot  the  root.    Having  made  theac  addirioni, 

a^  in   i  that  the  |  >  i  1  *  -  doOl  DOfl    retain  the  cubic  form, 

n,  hi,  and/;/,  being  vacant     Bach  of  tie 

:ig,   S   inc.  and  2  inches   thick;  i.e.   th* 


288 


EVOLUTION. 


area  covoml  to  tin-  depth  of  two  ndm  by  filling  the  n 

Ben  in  Fig.  2,  as  seen  in  Fig.  3,  is  2  X  40  X  ,VJ  =  -^n  square 

inches  ;  and  still  there  is  a  vacant  corner  n,  n,  n,  as  seen  in 


40  2 

which  is  a  cube  of  2  inches  on  each  edge ;  Le.  it  is  a  solid  2 

Fig.  1. 


4<» 


z/~ 


40 


71 


40 


40 


beta  thick,  (the  common  thickness  of  all  the  additions),  covering 
2X2  =  4  square  inches,  as  seen  in  Fig.  4. 


rrr.E   ROOT. 


280 


Now,  if  the  several  additions  made  in  Figs.  2,  3,  and  4,  be 
spread  out  upon  a  plane,  as  in 

Fig.  5, 
/ 


7U 


7\  £ 


7\£ZA<£Z?[ZZA 


\& 


or,  in  a  consolidated  form,  as  in 

Fig.  6, 


/  /  /    A 


3 


it  will  be  readily  seen  that  their  collective  solidity  will  be 
obtained    by  multiplying    the    entire    area    which    they    cover, 

(40  X  40  X  3  +  40  X  2  X  8  +  2  X  2  =  5044  gq»are 
inches),  by  their  common  thickness,  2,  which  will  give  10088 
solid  inches;  .\  a  cube  is  formed  (Fig.  4)  whose  edge  is  40  — |— 
2  =  42  inches,  and  no  blocks  remain. 

36.1.    If  there  are  more  than  two  figures  in  the  root,  the 
same  relations  subsist,  and  the  same  reasoning  applies.     Hence, 
To  extract  the  Cube  Root  of  a  Number, 

Kile.  1.  Separate  the  number  into  periods  of  three  figures 
each  by  setting  a  dot  over  units,  thousands,  etc. 

2.  Find  by  trial  the  greatest  cube  in  the  left-hand  period,  plans 
its  root  as  in  square  root,  subtract  the  cube  from  the  left-hand 
period  and  to  the  remainder  annex  the  next  period  for  a  divi- 
dend. 

Z.  Square  the  root  figure,  annex  two  ciphers  and  multiply  this 
result  by  3  for  a  tkiai.  divisor  ;  divide  the  dividend  by  the  trial 
divisor  and  set  the  quotient  as  the  next  figure  of  the  root. 


Rale  for  extracting  the  cube  root  of  a  number  ? 

u 


290  EVOLUTION. 

4.  Multiply  this  root  figure  by  the  part  of  the  root  previously 
obtained^  annex  one  cipher  and  multiply  this  result  by  three  ;  add 
the  last  product  and  the  square  of  the  last  root  figure  to  the  trial 

■  r,  and  the  jiM  will  be  the  TRUE  DIVISOR. 

5.  Multiply  the  true  divisor  by  the  last  root  figure,  subtract  the 
product  from  the  dividend,  and  to  the  remainder  annex  the  next 

I  for  a  new  dividend. 
G.  Find  a  new  trial  divisor,  and  proceed  as  before,  until  all  the 
periods  have  been  employed. 

Note  1.  The  notes  in  Ait.  357,  with  slight  modifications,  are  equally 
applicable  here. 

;  k  2.  If  the  root  consists  of  Tnree  figures  it  is  plain  that  the  cube,  as 
completed  in  Fig.  4,  must  be  enlarged  just  as  Fig.  1  has  already  been  enlarged. 
Hence,  the  new  trial  divisor  will  consist  of  3  faces  of  1%.  4  ;  lut  the 
true  divisor  already  fotuui  is  the  sura  of  the  significant  figures  in  these  3 
faces,  except  one  face  each  of  rr,  xx,  and  »,  and  two  faces  of  the  little  cor- 
ner cube,  nan  ;  moreover,  the  number  directly  above  the  true  divisor  (in 
the  operation)  represent!  am  face  of  win,  and  the  number  above  that  repre- 
sents the  sum  of  one  /ace  each  of  the  3  long  corner  blocks,  rr,  rr,  and  n ; 
hence,  tojind  the  next  trial  divisor,  we  have  only  to  add  the  true  divisor  alrradj 
/bund  to  twice  the  number  above  it,  and  oxck  the  number  above  that,  and  to 
ike  sum  annex  two  cipher*.  When  there  are  many  root  figures  this  process 
is  shorter  than  to  square  so  much  of  the  root  as  has  been  found,  annex  two 
ciphers,  and  multiply  by  3,  as  directed  in  the  3d  paragraph  of  the  rule. 

I  What  is  the  cube  root  of  21- 

21024576(276,  Ans. 
;<or=20»x3=       1200     8 
20  X  7  X  3  =         420 
7»=       _49 

\<i  Trir  Divisor  =       1669  13024 

2d  Trial  I)ivi,or  =  270*  X  3=  218700  11683 
■  ■  X  6  X  3=      4860 
6»  =  36 


2d  True  Divisor  =  2: 


19  11  .".76  2d  Dividend. 
1341576 


MM  k  contained  10  times  in  the  dividend,  yet 
the  root  figure  Ei  only  7.     The  true  root  figur-  \<  eed 

I  all  cases  be  found  by  trial. 
Squaring  2c  QM  result  as  squaring  2  and  annexing 

00,  as  directed  in  the  rule,  3d  paragraph. 


201 
8.  What  is  die  cube  root  of  G7917.: 


480000 
9C00 

64 

489664 


679  1  73  12(4  08,  Ans. 
G4 


3917 
3917312 


In  this  example,  the  1st  trial  divisor,  4800,  i-  larger  than  the 
1-t  dividend,  .;c.>17  j  .-.  we  annex  0  to  the  root,  00  to  the  1st  trial 
divisor  for  the  2d  trial  divisor,  tnd  bring  down  the  next  period 

:ii|.l<t«'  :i   new  dividend.     Tlie  rule,  followed  literally,  will 

give  the  same  result 

:  i:  3.    Prepare  fractions  and  mixed  numbers  as  directed  in  square  root 
s:>9). 

What  [|  the  value  of  the  following  expressions  : 

4.  V**0*3H  ?      Ana,  1 11.  11.  V;i,;-:,-,;,,;;7?     Ans.  3.33. 

5.  V317G^3?  12.  V10077-( 

6.  V382657176?  13.  VW.353607? 

7.  V^24024008?  11.  V1G6J?  Ans.  5}. 

8.  V387420489  ?  15.  V561gi  ? 

9.  V13 1217728?  16.  VH*  Ans.  1.65+. 
10.  V^  ?            Ans.  1.709  +.  17.  V*3f  ? 

Application  of  the  Cube  Root. 

360.  Bodies  which  are  of  precisely  the  same  form  are  simi- 
lar to  each  other,  and  the  solid  contents  of  similar  bodies  are  to 
each  other  as  the  cubes  of  their  corresponding  lines,  and  con- 
v<  r-<  lv,  the  corresponding  lines  are  to  each  other  as  the  cube  roots 
of  the  contents. 

Ex.  1.  If  an  iron  hall  I  inches  in  diameter  weighs  16  pounds, 
what  i-  the  weight  of  a  ball  30  inches  in  diam< 

it*,  or  l1  :6s::  16:  Ans.;    i.e.    1  :  216:: 

16Uk  :  ;;i."m;i!,..  a.,, 

365.    What   in  Note  1'     Note  2'     Explain   Ex.  2.     Ex.  3.     What   it  Note  8! 
366.  What  are  similar  Unties?    The  ratio  of  the  content*  of  similar  bodies! 


202  ARITHMETICAL   PROGRESSION. 

2.  If  a  bull  0  inches  in  diameter  weighs  27  pounds,  what  is 
the  diameter  of  a  ball  thai  freight  04  pounds? 

V27  :  VG  *  :  :  r'In-  :  Am.  :   i.  «  .  8  :   1  : :  Gin.  :  8in.,  An<=. 

3.  How  many  bullet*  \  of  an  inch  in  diameter  will  be  required 
to  make  a  ball  1  inch  in  diameter? 

1.  If  a  globe  of  gold  1  inch  in  diameter  tfl  worth  $100,  what 
is  the  diameter  of  a  globe  worth  $6400  ? 

5.  S ;ij i] the  diameter  of  the  earth  is  7012  miles,  and  that 

it  takes  1404928  bodies  like  the  earth  to  make  one  as  large  as 
the  sun,  what  is  the  diameter  of  the  sun? 

6.  A  bin  is  8  feet  long,  4  feet  wide,  and  2  feet  deep ;  what  is 
the  edge  of  a  cubical  box  that  will  hold  the  same  quantity  of 
grain  ? 

7.  If  a  stack  of  hay  2  1  foei  high  weighs  27  tons,  what  i<  the 
bight  of  a  similar  stark  which  weighs  8  to:  An*.    1 

8.  If  a  hell  1  inches  high,  3  inches  in  diameter,  and  \  of  an 
inch  thick  weighs  lib.,  what  are  the  dimensions  of  a  similar  bell 
that  weight  271b.? 

9.  If  a  loaf  of  sugar  10  inches  high  weighs  81b.,  what  is  the 
Light  of  a  similar  loaf  weighing  1  lb.  ? 


ARITHMETICAL    PROGRESSION. 

367.    Any  series  of  numbers  increasing  or  decreasing  by  a 

common  difference  is  in  Ai:i  i hmf.tk  ai.   Progression; 
thus,     2,     5,     8,  11,  14,  17,  etc.  is  an  ascending  series, 
and     35,  30,  25,  20,  15,  10,  etc  is  a  descending  sen 
The  several  numbers  forming  a  series  are  called  Terms  ;  the 

fir»t   and    last   terms,  Extremes;    the   others,  Means.      The 

ditVerence  between  two  successive  terms  is  the  Common  Dif- 

i  iiiiNCE. 

367.    When  is  a  series  of  numbers  in  Arithmetical  Trogreasion?    How  many 
kinds  of  series?     What?     What  ara  the.  Terms  of  a  series ? 


ARITHMETICAL   1  293 

In  an  arithmetical  alars  ekrii  dHoo, 

viz.  tln>  fir-t  term,  last  term,  common  dUfereoce, nomber of  terms, 

and  sum  of  all  tho  term-;   and  tin  related  to  each  other 

that  If  any  three  of  them  are  given  the  other  two  can  be  found. 

3GS.    in  an  ascendin  let  6  be  the  first  term  and  5 

the  common  diflereo 

Then  6  =  1  -t    term. 

C  +  5  =  ll=2d   term, 
C-f-5  +  5  =  6  +  2x5  =  16  =  3d    term. 
C  +  5  +  54-5  =  6  +  3x5  =  21  =  4th  term. 
Tin:  that,  in  an  ascending  series,  the  second  term  is 

found  by  adding  the  common  difference  once  to  the  first  term  ; 
the  tliir.l  term,  by  adding  the  common  difference  twice  to  the 
Jirst  term,  etc. 

A   similar   explanation    may   be    given   when    the    ser; 
descending.     Hence, 

369.  Problem  1.  To  find  the  last  term,  the  first 
term,  common  difference,  and  number  of  terms  being 
given  : 

Rule.  Multiply  the  common  difference  by  the  number  of 
terms  less  1  /  add  the  product  to  the  Jirst  term  if  the  series  is 
ascending,  or  subtract  the  product  from  the  first  term  if  the 
series  is  descending,  and  the  sum  or  difference  will  be  the  term 
sought. 

.   1.  If  the  first  term  of  an  ascending  5,  the  com- 

mon difference  4,  and  the  number  of  terms  7,  what  is  ths 
term  ?  5  +  6  X  4  =  29,  Ans. 

%  The  first  term  of  a  descending  series  is  47  and  the  com- 
mon difference  8  ;  what  is  the  Cth  term  ? 

•17  —  5X8  =  7,.' 
3.  What  il  the  amount  of  $100,  at  G  per  cent.,  simple  inter- 
.:- ': 

307.  What  are  the  Extremes  of  a  series?  Means?  Common  Difference'  11  \r 
many  particulars  claim  special  attention'  What  ure  thev ■?  llow  many  of  them 
must  be  given?  368.  llow  is  an  ascending  series  formed  ?  How  a  descend* 
ing  series?    309.    Object  of  Problem  1 '    liule? 

25« 


294  ARITIIMKTK  AL    PROGRESS  I 

370.  Problem  2.  Tu  Snd  the  common  difference, 
the  extremes  mid  Dumber  of  terms  being  iri\ 

By  inspecting  the  formation  of  the  series  in  Art.  3C^,  it  will 
be  seen  that  the  difference '  between  the  extremes  is  equal  t 
common  difference  multiplied  by  1  less  than  the  number  of  terms  ; 
e.g.  the  difference  betWW  n  tin-  1st  and  4tli  term  (91  — C  — 
is  the  sum  of  3  equal  addition-  ;  .-.  this  difference,  divided 
(16  -$-8  =  5),  gives  one  of  these  additions,  i.  e.  the  common 
ference.     Hence, 

lit  if.  Divide  the  difference  of  the  extremes  by  the  number 
of  terms  less  one,  and  the  quotient  will  be  the  comtn 

Ex.  1.  The  extremes  of  an  arithmetical  series  are  3  and  38, 
and  the  number  of  terms  is  8  ;  what  is  the  common  difference  ? 

38  —  3  __  35 

2.  A  man  has  6  sons  whose  ages  form  an  arithmetical  set 
the  youngest  is  2  years  old  and  the  oldest  22  ;  what  is  the  dif- 
fen not  Of  their  a_  Ans.  4 

3.  The  amount  of  SI 00  at  simple  interest  for  10  years  is  $1G0  ; 
what  is  the  rate  per  cen: 

371.  Problems.  To  find  the  (lumber  of  terms,  the 
extremes  and  common  difference  being  gi\ 

By  Art.  MB  it  i-  •\i.lnit  that  the  difference  of  tin-  extremes 
is  the  common  difference  multiplied  by  one  leas  than  the  number 

rms.      11  lv, 

Kt  i  i  .  1 >  ride  the  difference  of  the  extremes  by  the  common 
difference,  and  the  quotient,  increased  by  1,  is  the  number  of 
terms. 

i.x.  1.  The  extremes  of  an  arithmetical  series  are  3  and  31 

and  the  common  difference  is  4;  whit  b  the  number  of  terms? 

31— 3    ,    ,       28 


+  1=T+1=7  +  1=8, 


4 

2.  The  common  difference  in  the  ages  of  the  children  in  a 
family  is  2  petti]  the  youngest  is  1  year  old  and  the  oldest  l'J; 
how  many  children  in  the  family  ? 


Glomeruli' a l  ii  295 

372.   Pbobhm  t.    To  fM  fee  quo  of  q  • 
extremes  and  number  of  terms  l 

>uin  of  the   i  qoaJ  to  th<-   tum  of  any  two 

equally  distant   from   tl  .:;   the 

•,11.  i:;.  we  1 
]  m  -f-  Oth  =  2d  +  5th  =  3d  -4-  4th. 
3  +  13  =  5  +  H  s=  7  -f  9  =16) 
.  the  ram  of  nil  terms  ii  10  x  3  =  48.     1  :• 

BlTLB.      Multiply  tkt  sum  "J   the   extremes  by  half  the   nn- 
of  terms,  ami  the  product  is  the  sum  of  the  series. 

Ex.  1.   The  e  4  a  leftSS  MM  •">  and  .')■.'  and  the  numl»«T 

of  terms  is  10 j   what  i>  (hi  MUl  of  the  series? 

3  +  39  =  42  ;  10  -*-  2  =  5  ;  42  X  5  =  210,  Ans. 

2.  How  many  strokes  does  a  clock  strike  in  12  hours? 


GEOMETRICAL   PROGRESSION. 

373.  Any  series  of  numbers  increasing  or  decreasing 
by  a  common  ratio  is  in  Geometuical  Puoguession  ; 

tlniS,     2,      0,   18.  .">!,    162,  etc.  is  an  ascend  in. 
and     CI,  32,  16,    8,       1.  etc.  is  a  desoeodiii 

In  tin-  above,  8  is  the  ratio  in  the  1-t  .-eric-  and  I  in  the  2d. 

The  first  tana,  last  term,  ratio,  number  of  terms,  and  sum  of 
all  the  terms  are  so  related  to  each  other  that  if  any  three  of  them 
are  given  the  other  two  can  be  found. 

374.  In  a  series,  let  2  be  the  first  term,  and  4  the  ratio ; 
Then  2  =  1st  term. 

2X4=      8  =  2d   term. 

2X4X4  =  2X4*=    32  =  N  torn 

2X4X4X4  =  2X4S=128=  4th  tens. 

370.     Object    of   Problem    2?      Rule?      371.     Object    of   Problem    3' 

i'    Kuk  '    373.    What  constitute*  a  aeries  in  Ceometri- 
Of  *erie«<?    What?    How  many  particular! 
claim  attcntiou!    WSttf     How  many 


296  GEOMETRICAL  PROGRESSION. 

In   forming  the  I  M  we  see  that  the  second  term  13 

found  by  multiplying  the  jirst  term  by  the  ratio  ;  the  third  term, 
by  multiplying  the  firU  by  the  square  of  the  ratio;  the  fourth,  l»y 
multiplying  the  Jirst  by  the  cube  of  the  ratio,  Me  index  of  the 
power  of  the  ratio  always  being  one  less  than  the  number  of 
the  term  sought.     A  similar  explanation  may  .  when  the 

scries  is  descending.     Hence, 

375.  Problem  1.  Fo  find  the  last  terra,  the  first 
I  mi,  ratio,  and  number  of  terms  being  given, 

Rule.  Multiply  the  first  term  by  that  power  of  the  ratio 
whose  index  is  equal  to  the  number  of  terms  preceding  the  required 
term,  and  the  product  will  be  the  term  sought. 

Ex.  1.  The  first  term  of  a  geometrical  series  is  4,  the  ratio  3, 
and  the  number  of  terms  8]  what  is  the  last  term? 

C  —  1  =  5 ;  3*  =  148  ;  tad  148  X  4=  972,  An<. 

2.  The  1st  term  is  3,  and  the  ratio  £;  what  is  the  5th  terra? 

5  —  1  =  4;  (1)4  =  A;  and^  X3  =  ^,  Ans. 

3.  The  1st  term  is  5,  the  ratio  1.06 ;  what  is  the  4th  tens  ? 

is.  5.95508. 

4.  What  is  the  amount  of  $10  at  compound  interest  for  4 
years  at  5  per  cent,  per  annum? 

5.  Supposing  money  at  compound  interest  to  double  once  in 
12  years,  to  what  will  $100  amount  in  72  years? 

.   $f.400. 

376.  Since  the  last  term  is  obtained  (Art.  374)  by  multiply- 
ing the  first  term  by  that  power  of  the  ratio  whose  index  is  equal 
to  the  number  of  terms  less  one,  so,  conver 

PftOBLBM  2.  To  find  the  ratio,  the  extremes  and  num- 
ber of  terms  being  given  : 

Rule.  Divide  the  last  term  by  the  first,  and  the  quotient 
will  be  that  power  of  the  ratio  whose  index  is  one  less  than  the 
number  of  terms ;  the  corresponding  root  of  the  quotient  will 
therefore  be  the  ratio. 

374.  How  is  an  ascending  series  formed?    A  descending  series?    375.  Object 
of  Problem  1?     Rule?     376.   Object  of  Problem  2?    Bole? 


GE  \. 

..  1.  The  firefl  torn  in  i  geonetrieal  toriei  ii  2,  the  la>t  term 
250,  and  the  number  of  term*  1  ;  what  Ml  the  ratio? 
j  8 -a  IK,  -i  —  i  = ;; ;  tod   N' 

2.  The  i  Sand   l*.  and  the  nnmhnr  of  Ur—  fl  j 

what  il  the  ratio?  .  4  or  |. 

mes  are  3  and  243,  ami  the  number  of  terms  5; 
what  is  the  ra: 

377.    Problem  3.     To  find  the  mm  of  a  M 

extreme's  ami  ratio  being  given. 

Having  a  Beriee  given,  e,  g.  2,  10,  50,  MO,  1250,  C2."><>,  wu/- 
://)///  each  trnn  except  the  last  by  the  rutin,  5  ;  t! 

2,     10,      50,      M0,     l  M0,     [6960], 

Produet   i  10,       50,       250,     U  '■<) ; 

and  we.  shall  evidently  form  a  ftftp  I  Ties  like  the  old,  exe.-pt  the 
lir.-t   term  of  the  pJU  <aml   in   the   new.      Now,  if  li. 

except  the  last  term  be  subtracted  from  the  >"  "'.  the  remaindrr  tW// 
ta  {/«"  0  o/  fta  extremes  in  the  old  series  the  other  terras 

in  the  two  ich  Other;   the   remainder  w;'.. 

be  4  times  the  sum  of  all  the  terms  except  the  last  in  the  old 
-  ;  for  once  a  leriei  from  8  nfwm  a  series  must  h 
BfietJ   ."•   |  of  this   remaindeJ    plus  the  lust   term  must  be  the 
sum  of  all  the  terms  in  the  old  series  ;   but  4  is  the  ratio  less  1. 

A  similar  explanation  is  always  applicable.      Hence. 

Kile.     Divide   the   difference  of  the   extremes  by  the  ratio 
less  one,  and  to  the  quutitnt  add  the  greater  < 

Ex.  L  The  extreme,  are  '2  and  486,  and  the  ratio  3  ;  what  is 
the  sum  of  the  series  ? 

486  —  2  =  484  ;  3—1  =  2;  484  -f-  2  =  242  ;  and  212  -f 
480  =  728,  Ana. 

mes  are  4  and  518  1,  and  the  ratio  0  ;  what  is  the 
sum  of  the  sere 

3.  What  debt  will  be  diecharfed  by  12  monthlr payments, 

1-t   payment  being   $1,  the  2d   $2.  and  po  on  in   i  geometrical 


37  7.    <  'bjcet  of  l'roblim  I  I 


298  ANNUITIES. 


ANNUITI 


378.  An  Annuity  is  a  sura  of  money  payable  annually,  or 
at  any  regular  period,  either  for  a  limited  time  or  forever. 

An  annuity  is  in  arrears  when  the  installments  remain  unpaid 
after  they  are  due. 

The  Amount  of  an  annuity  in  arrears  is  the  interest  of  the 
unpaid  installments  added  to  their  sum. 

379.  Problem  1.  To  find  the  amount  of  an  annuity 
in  arrears,  at  simple  interest. 

Ex.  1.  An  annuity  of  $100  per  annum  has  remained  unpaid 
4  years;  what  is  its  amount?  Ans.  $436. 

The  4th  payment  is  due  to-day  and  is  worth  just  $100;  the 
3d  payment  due  1  year  ago  i>  worth  $106  ;  the  2d  payment  due 
2  years  ago  ifl  worth  $112  j  and  the  1st  payment  due  3  years 
ago  is  worth  $118.  But  these  numbers.  $100,  $106,  $112,  and 
$118,  are  in  arithmetical  progression.     Hence, 

BULB.  Fmd  the  last  term  of  the  series  by  Art.  369-,  and  the 
sum  of  the  series  by  Art.  372. 

2.  Purchased  B  farm  for  $5000,  agreeing  to  pay  for  it  in  5 
equal  annual  installments ;  the  5  years  having  elapsed  without 
any  payment  being  made,  what  is  now  due,  allowing  simple 
interest?  Ans.  $5600. 

3.  A  salary  of  $600  per  annum  is  in  arrears  for  8  years ;  to 
what  does  it  amount,  allowing  simple  interest  at  7  per  cent.  ? 

380.  Problem  2.  To  find  the  amount  of  an  annuity 
in  arrears  at  compound  interest. 

Ex.  1.  What  is  the  amount  of  $1  annuity,  per  annum,  in 
arrears  for  3  years,  at  6  per  cent,  compound  interest  ? 

The  3d  instalment  becoming  due  to-day,  is  worth  just  $1  ;  the 
2d  having  been  due  1  year,  is  worth  $1.06;  and  the  1st  having 

378.    What  is  an  Annuity  ?    When  is  an   annuity  in   arrears  ?    What  is  the 
Amoant  of  an  annuity?    3T9.    Object  of  Problem  1 7    Rule?    3S0.    rrot>:eui2? 


. 


299 


due  2  years,  is  worth  $1  $1  -f$1.06  +  $l 

r=$3.133t*>,  the  >um  sought.     Hut  these  uumbON  are  in  geomet- 
rical progression.     Hence, 

Ki  i  |  1.      Find  the  hist  term  of  the  series  by  Ar  <l  the 

sum  of  the  series  by  Art.  377  ;  or, 

Rci  I  -.      Multiply  the  amount  of  $1,  found  in  the 
,  by  the  annuity,  and  the  product  will  be  the  required  amount. 

TAB 

Showing  tht  amount  of  the  annuity  of  $\,  £\,  etc ,  at  4,  5,  6,  and  7  />< 
compound  interest,  from  1  to  20  yean. 


V.arv 

4  p«r  Cent. 

6  per  Cent. 

7  per  1 

Ye»n. 

1 

1.000000 

1  000000 

1.000000 

1.000000 

1 

2 

2040000 

2  050000 

2.060000 

2.070000 

2 

3 

3.121600 

3.152500 

a.  ii 

3.214900 

3 

4 

4.3101  2  j 

4.374616 

4.4  : 

4 

5 

5.41' 

15581 

<»739 

5 

6 

6.632975 

6.801913 

6975319 

7  18 

6 

7 

7.89- 

8.142008 

•5838 

:o21 

7 

8 

921  i 

9.549109 

•7468 

8 

9 

10.562795 

11.0* 

11.491316 

11. '.i7 

9 

10 

.,107 

12.577 

18.151 

15  61 

10 

11 

13.486351 

14.206787 

14.971645 

18  788589 

11 

12 

15.085806 

15.9171-7 

16  869941 

17.888451 

12 

13 

16.626838 

17  712983 

18.882138 

80.140548 

13 

14 

18.291')  11 

19.598632 

81.01 

0488 

14 

15 

20.023588 

21.578564 

25.129022 

15 

16 

21.824531 

88.853 

27.888054 

16 

17 

23.697  :.12 

25.840366 

2880 

10817 

17 

18 

85.545419 

28.13  <85 

19039 

18 

L9 

27.671229 

»992 

37  :i7S965 

19 

20 

S079 

33.065  4 

88.788 

40.995492 

20 

2.  What   is    the   amount  of    an   annual   salary  of   $1000,   iu 
■mm  l\>r  it  6  per  cent.?  m,  $5637.093. 

3.  What  is  the  amount  of  an  annual  rent  of  100£,  in  arrears 
for  15  years,  at  5  per  cent.  ? 

Ans.  2157.85G4£  =  2157£  17s.  Id.  2qr. 
1.   What   i>   the  amount  of  an   annual  pension  of   $500,  in 
arrears  for  12  years,  at  6  per  cent.? 


lit  EuU?    Id  Bul.r 


PERMUTATIONS 


381.  Pi-KMii  ation  it  tin-  arranging  of  a  given  number  of 
things  in  every  possible  order  of  miOOfionloil 

382.  Problem.  To  find  the  number  of  permutations 
of  a  given  number  of  thin 

The  single  letter,  a,  can  have  but  1  position,  i.e.  it  cannot 
ftand  either  before  or  after  itself;  the  2  letters,  a  and  b,  furnish 
the  *J  permutation-, 

■]  ?     >■ ,  the  number  of  which  is  expressed  by  the  product  of 

1  X  2  =  2;  and  if  a  3d   letter,   c,  be   introduced,  we  have 
(c  a  by  c  b  a) 

<acb,  b  c  a  [  ;  i.  e.  the  new  letter,  c,  may  stand  1st,  2d,  or  3d 
(a  b  c,  b  a  c) 

in  Men  of  the  I  permutations  of  a  and  b  ;  hence  the  number  of 

permutations  of  3  thing-  II  expressed  by  the  product,  1  X  -  X  3 

=  6.     If  a  4th  tetter,  d.  be  taken,  it  may  stand  as  1st,  2d,  3d,  or 

4th,  in  each  of  the  6  permutations  of  a,  b,  and  c,  and,  of  course, 

faraiea  I  lanei  6=1x2x3x4  =  24  permutation*. 

By  the  above,  it  is  evident  that  the  number  of  permutations 

Of  1  thing  =      1 

Of  I  thing*  =  1X2=     2 

Of  3  thing*  =  1X2X3=      6 

Of  4  things  =1X2X3X4=    24 

and  so  on  to  any  extent.     Heaee, 

BULK.  Form  the  series  of  numbers,  1,  2,  3,  4,  etc.,  up  to  tfo 
number  of  things  to  be  permuted,  and  their  continued  product  will 
be  the  number  of  permutations. 

Ex.  1.  How  many  different   integral   numbers   may  be   ex- 
piv-sed  by  writing  the  9   significant  digit*  in   sueeession,   eaeh 
figure  to  be  taken  once,  and  but  once,  in  each  number? 
Ans.  1X2X3X4X5X6X7X8  X  9  =  362880. 
2.  In  how  many  different  orders  may  a  family  of  10  persons 
seat  themselves  around  the  tea  table  ? 

381.    What  i«  Permutation?    383.    Object  of  the  Problem?    Rulef 


MENSURATION. 

383.    Mensuration  is  the  art  of  i  lines,  surfaces, 

and  solid<. 

'I'lic   principle!  are  all  (ieomrtri.al,  and   are   very   iiuii). 
A  few  only  of  the  more  simple  are  here  present < d. 

3>  1.     Two   parallel    line-  ere      

equally  distant  from  each  other.  

When  two  lines  meet  so  as  to  form  * 

angles,   the   lims    an-  perpendicular   to 
Other  and  the  angh a  an    right  angles.     A 
right  angle  contains  90°. 

An  acute  angle  is  an    angle  of  less  than 
90°. 

An  obtuse  angle  is  an  angle  of  more    than 
90°. 

Two  lines  are  oblique  to  each  other   whoa 
the j  meet  so  as  to  form  acute  or  obtuse  angles,  and  the  a 
are  oblique  angles. 

385.    A  Triangle  is  a  plane  figure  which 
is  bounded  by  three  Hoes. 

Tlie  base  of  a  triangle  (or  any  other  figun  ) 
i-   the   side    on    which    it    i<    MppOOed    to    >tand. 
altitude  of  a  triangle  is  tlie  ncfpnndinnhir 
d'wtance  from  the  ungle  opposite  the  1.  to  the 

led. 
380.    PbOjUM  1.    To  find  the  area  of  a  tri 

lit  ii;.     Multiply  the  base  by  half  the  altitude. 
Bat  1.     Tlie  base  ol  a  triangle   is   7   inches  and   the  altitude 
8  inches  ;  what  i<  its  ari  ;.  in. 

J.   The  hase  is  8ft.  and  the  hight  lift.;   what  i-  the  area? 

883.    What  is  Mensuration?    384.    What  of  two  parallel  line**    What  it  a 
right  angle?    An  acute  angle?    Obtuse  angle?    What  are  oblique  line*?    0 
angle*?    3*3.    What  is  a  Triangle?     IU  baae?    lu  alUtude?     380.    Bule  for 
^   it*  area* 

2G 


302  MENSURATION. 

387.     A   Quadrilateral   or  Quadrangle   is   a  plan^ 
figure,  having  four  sides  and  four  angles. 

There  are  three  kinds  of  quadrilaterals,  viz. : 

1st.    Trapeziums,   none  of  whose  sidej    Are 
parallel ; 


2d.    Trapezoids,  as  A  B  C  D,  only  one 
pair  of  whose  sides  are  parallel ;  and, 


A  E 


3d.    Parallelograms,  each  pair  of 
whose  opposite  sides  are  parallel,  as 
A  B  C  D,  or  F  E  C  D. 
A       F  BE 

The  diagonal  of  a  figure  is  a  line  which  joins  two  opposite 
angles,  as  A  C  in  the  above  trapezoid,  and  B  D  in  the  parallelo- 
gram. The  altitude  of  a  trapezoid  or  parallelogram  is  the  per- 
pendicular between  two  parallel  lidet. 

388.  Problem  2.    To  find  the  area  of  a  trapezium  : 

Rule.  Draw  a  diagonal  dividing  the  trapezium  into  tiro 
triangles,  and  find  the  area  of  each  triangle  by  Problem  1.  The 
sum  of  these  triangles  will  be  the  area  of  the  trapezium. 

Ex.  What  jl  the  area  of  a  trapezium,  one  of  who*e  diagonals 
is  20  inches,  and  the  length  of  the  perpendiculars  let  fall  upon 
it,  from  the  other  angles  of  the  trapezium,  G  and  8  inches? 

Ans.  140sq.  in. 

389.  Problem  3.    To  find  the  area  of  a  trapezoid  : 

Rule.  Multiply  the  half  sum  of  the  parallel  sides  by  the  alti- 
tude, and  the  product  will  be  the  area. 

387.  What  in  a  Quadrilateral?  How  many  kinds?  What  is  a  trapezium? 
Trapezoid?  Parallelogram?  What  ia  the  diagonal  of  a  figure?  Altitude  of  a 
trapezoid?  Of  a  parallelogram?  388.  Rule  for  finding  the  area  of  a  trapezium? 
3S9.    Rule  for  finding  the  area  of  a  trapezoid? 


MENSURATI'  M 

Ex.  1.    The  parallel  sides  of  a  trapezoid  are  10  and  II 
and  its  altitude  its  area?  j.ft. 

2.   What    Kl   Um  ar« «  of  a  board,  whose   length   is    1<M:.. 
wider  end  being  2ft.  and  the  narrower  1  -  in  width  ? 

JI90.  PftOBUM  4.  To  find  the  area  of  a  parallelo- 
gram : 

Ki  i.e.  Multiply  the  base  by  the  altitude,  and  the  product  is 
the  area. 

.1.    What  is  the  area  of  a  rectangular  field,  whose  1- 
is  40  rods,  and  altitude  or  width  8  rods  ? 

2.  The  base  of  a  parallelogram  is  C  feet,  and  the  altitude  4 
what  i>  its  area  ? 

391.  A  Polygon  is  a  plain  figure  bounded  by  straight 
lines. 

ml.    Three  straight  lines,  at  least,  are  required  to  bound  a  polygon. 

The  lines  which  bound  a  polygon,  taken  together,  are  called 
the  perimeter  of  the  polygon. 

A  polygon  of  5  sides  is  called  a  pentagon  ;  of  6,  a  hexagon  ;  7, 
a  heptagon  ;  8,  an  octagon  ;  9,  a  nonagon  ;  10,  a  decagon  ;  11, 
an  undecagon  ;   1 2,  a  dodecagon  ;  etc. 

Note  2.  A  polygon  may  he  divided  into  triangles  by  drawing  diagonals, 
and  then  its  area  may  be  found  by  Problem  1. 

399.  Problem  5.  To  find  the  area  of  a  circle  when 
tho  radius  and  circumference  are  given  (Art.  109  and 
361): 

Rile   1.     Multiply  the  circumference  by  half  the  radius;  or, 

Rule  2.  Multiply  the  square  of  the  radius  by  3.141592,  and 
the  product  is  the  area. 

1.  What  is  the  area  of  a  circle,  whose  radius  is  C  and  cir- 
cumference 37.699104?  Ans.  113.097 

2.  What  is  the  area  of  a  circle  whose  radius  is  10? 

390.    Rufe  for  finding  the  are*  of  a  parallelogram?    391.    What  U  a  Polygon? 
miter  of  a  polygon?    Name  the  different  polygons?    393.    Rale  for 
finding  the  area  of  a  circle?    Second  Rale? 


30-4  MENSURATION. 

393.  A  Prism  is  a  solid  that  has  two 
similar,  equal,  parallel  faces,  called  bases, 
and  all  its  other  faces  parallelograms. 

Note.  A  prism  is  triangular,  quadrangular,  pentagonal,  etc.,  according 
as  its  bases  are  triangles,  quadrangles,  peutagons,  etc. 

A  Cylinder  is  a  round  body  whose 
diameter  is  the  same  throughout  its  entire 
length,  and  whose  ends  or  bases  are  equal, 
parallel  circles. 

394.  Problem  G.  To  find  the  surface  of  a  prism  or 
cylinder  : 

RULE.  Multiply  the  perimeter  or  circumference  of  the  base 
by  the  length  of  the  solid,  and  to  the  product  add  the  area  of  the 
two  ends. 

Ex.  1.  What  is  the  surface  of  a  prism,  whose  length  is  10 
inchi's  and  l>a~<-  \  inches  squan  ?  Ans.  L92#q.in. 

2.  What  ifl  the  surface  of  a  cylinder,  whose  length  is  80 
and  diam  at? 

395.  Problem  7.  To  find  the  solid  contents  of  a 
prism  or  cylinder : 

K    ii.      Multiply  the  area  of  the  base  by  the  altitude. 

V.\.  1.  What  are  the  contents  of  a  cylinder,  whose  length  is 
20imli< %  and  whose  diameter  is  10  inches? 

An<.    l.-»7<>.7'JGc.  in. 

2.  What  are  the  contents  of  a  quadrangular  prism,  whoso 
length  is  2b  feel  and  whose  base  is  3  feet  square  ? 

396.    A  Pyramid  is  a  solid,  having  a  polygonal 

face,  called  the  base,  and  all  its  other  faces  are  trian- 

.  hich  meet  at  a  common  point,  called  the  vertex 

of  the  pyramid.     The  slant  hight  is  the  distance  from 

the  vertex  to  the  middle  of  one  side  of  the  base. 

393.  What  is  a  Priam!  A  Cylinder?  394.  Rule  for  finding  the  surface  of 
a  prism  or  cylinder?  395.  Rule  for  finding  the  contents  of  a  prism  or  cylin- 
der?   390.    What  is  a  Tyramid?    Its  vertex?    Slant  hight? 


now.  805 

p..     A  pyramid  u  triangular,  quadrangular,  vt  _;  a*  iU  base 

ii  a  triangle,  quadrangle,  etc. 


A  I  ■  solid,  like  a  pyramid,  except  that  ita 

base  is  a  circle.     The  altitude  of  the  pyramid  or  cone 

.  crpemlicular  hiyht. 


397.  Problem  8.  To  find  the  contents  of  a  pyra- 
mid or  of  a  cone : 

Kile.  Multiply  the  area  of  the  base  by  one  third  of  the 
altitude. 

Ex.  1.  What  are  the  contents  of  a  cone,  whose  base  is  10 
feet  in  diameter  and  whose  altitude  is  24  fe. 

Ans.  G28.3184cu.n. 

2.  Whftl  are  the  contents  of  a  pyramid,  whose  altitude  ii  12 
Inches  and  whose  base  is  a  triangle,  having  its  base  G  inches  and 
its  altitude  8  inches  ? 

398.  The  Frustum  of  a  pyramid 
or  cone  is  the  part  remaining  after  a 
portion  next  the  vertex  has  been  cut  off 
by  a  plane  parallel  to  the  base.  The 
two  ends  are  called  the  upper  and  lower 
bases. 

399.  Problem  9.  To  find  the  contents  of  the  frus- 
tum of  a  pyramid  or  cone  : 

Krir.  Multiply  the  sum  of  the  ttco  bases,  added  to  the  mean 
proportional  between  the  two  bases,  by  one  third  of  the  altitude 
of  the  frustum. 

Ex.  1.  What  are  the  contents  of  the  frustum  of  a  quadrangu- 

390.  What  is  a  Cone?  Altitude  of  a  pyramid  or  cone?  397.  Rule  for  find- 
ing the  solid  contents?  399.  What  la  the  Frustum  of  a  pyramid  or  cone? 
899.    Content*  of  a  frustum,  how  found? 


30G  ■■■■»■  I  lf«> 

lar  pyramid,  whose  altitude  is  21  feet  and  whose  bases  are  5 
feel  ami  .'}  feet  square?  Ans.  3-i«'k*u,  ft, 

2.  What  are  the  contents  of  the  frustum  of  a  cone,  whose 
hijHit  is  12  feet  and  whose  bases  are  0  feet  and  4  feel  in  diam- 
eter  ? 

100.  A  Spiiere  or  Globe  is  a  solid 
bounded  by  a  curved  surface,  all  peril  of 
the  surface  b<iiiLr  equally  distant  from  a 
point  within,  calk-d  the  centi  r. 

A  diameter  of  the  sphere  is  a  Roe  patt- 
ing through  the  center,  and  limited  in  both 
directions  by  the  surfi 
401.     Problem  10.    To  find  the  surface  of  a  sphere  : 
Kile.     Multiply  the  circumference  by  the  diameter. 

Ex.  1.  What  is  the  surface  of  a  sphere,  whose  diameter  is 
100  inches?  An*  61  -H5.92>q.  in. 

2.  What  is  the  surface  of  the  earth,  supposing  it  to  be  a  sphere 
8000  miles  in  diameter? 

3.  What   is  the  surface  of   the  sun,  supposing  it  a  sphere 

diameter  is  885680  mi 

403.   Problem  11.    To  find  the  contents  of  a  sphere  : 

Rule  1.  Multiply  the  surface  of  the  sphere  by  one  third  of 
the  radius. 

Role  2.     Multiply  the  cube  of  the  diameter  by  the  decimal 
?99;  i.e.byl  of  3.141592. 

Ex.  1.  What  are  the  contents  of  a  sphere,  whose  diameter  is 
100  inches?  .  523598§c.iu. 

2.  What  ia  the  volume  or  solidity  of  the  earth,  supposing  it  a 
sphere  whose  diameter  is  8000  miles  ? 

3.  What  is  the  volume  or  solidity  of  the  sun,  supposing  it  a 
sphere  whose  diameter  is  885680  mil. 

400.  What  is  a  sphere?  Its  diameter?  401.  Rule  for  finding  the  surface  of 
a  sphere?  403.  Rule  for  finding  the  volume  or  solid  contents  of  a  sphere? 
Second  rule? 


KXAAIl'LES.  Ml 


MIm   I   LLANEOUS    EXAMPLES. 

1     What  number  Increased  t  _'0? 

2.  What  Dumber  diminished  by  1$  gives  21  t 

of  two  numb.  ;  d  0116  of  the  numb. 

I)  times  the  other;   what  an-  the  numb. 

1  Odi  and  tttfl  fOdi  are  what  part  of  an  acre? 

5.  The  difference   between   two  Demberi   i-  87| 
smaller  nember  if  12*;  what  is  the  lai 

a  hat  number  multiplied  by  86]  givm  1000? 

7.  What  Bember  divided  by  • 

8.  What  is  the  greatest  common  divisor  of  84  and  1 

'.'.    What  is  the  least  common  multiple  of  72  and  364? 

10.  Wbal  k  the  interest  of  9766.64  fcr  *m.  17 

11.  The  diflereooe  between  two  oamben  k  2$,  and  me  smaller 
number  is  10;  what  is  the  larger?     Whet  the  sum  of  th< 
numbers  ? 

1  2.  The  difference  of  two  numbers  is  563492,  and  the  larger 
number  is  .'3042.338  ;  what  is  the  smaller?  What  the  mm  of  the 
two  numbers  ?  let  Ana  8079046. 

13.  How  many  bricks  8  inches  long,  1  inches  wide,  and  2 
inches  thick,  will  be  required  to  build  I  wall  20  feet  long,  16 
feet  hi«rh,and  2\  feet  thick? 

1  1.    How  many  brick>  whole  dimensions  are  8',  4',  and  2',  will 
it   take*  to  build  the  walls  of  a  DOOM  40ft,  long,  28ft.  wid. 
22ft.  high,  the  walls  to  be  1ft.  6'  thick,  and  no  allowance  made 
for  doors  and  windows  ? 

1.").  The    salary   of  the    President   of  the    T'nitcd    State*  is 
")  per  annum;  what   sum   may  he  expend  daily,  and  yet 
560  in  one  term  of  office,  viz.  4  years?        Ans.  ^ 

1  6.    What  number,  multiplied  by  h  of  itself,  will  pro 

17.  What  number,  multiplied  by  •}  of  itself,  will   pfodnet 

18.  How  mai  will  it  take  to  lay  a  floor 
20ft.  long  and  16ft.  wide? 

How  large  a  square  floor  can  be  laid  with  676  6quar« 
i 


308  MISCELLANEOUS    EXAMPLES. 

20.  The  fore  wheel  of  a  carriage  ia  9  feet,  and  the  hind  wheel 
10^  feet  in  circumference j  how  many  times  will  each  turn  round 
in  running  from  Boston  to  Andover,  20^  mile-  ? 

21.  A  rectangular  piece  of  land,  containing  00  acres,  has  its 

length  to  its  breadth  as  3  to  2,  what  are  its  length  and  breadth? 

22.  Bought  a  cask  of  molasses,  containing  84  gallons,  for  J 
but  9  gallons  having  leaked  out,  at  what  price  per  gallon  must  I 
sell  the  remainder  to  gain  $1.2.;?  Ans.   48  cents. 

23.  If  a  pipe  G  inches  in  diameter  will  discharge  a  certain 
quantity  of  water  in  1  boors,  in  what  time  will  a  4-inch  pipe 
discharge  the  lantity?  .  1)  liours. 

2  1.    In  l2gaL  .""jt.  ipt  2gi.,  how  many  pills  ? 

2j.  In  1846542  seconds  how  many  week  etc? 

26.  Resolve  207  1( »  into  its  prime  factors. 

Ana.  2,  2,3,3,  5,  11,  13. 

27.  Reduce  ft,  ^3,  ft,  and  *V  to  equivalent  fractions  having 
the  least  common  denominator. 

28.  Reduce  3s.  4d.  2qr.  to  the  fraction  of  a  pound. 

29.  Reduce  ft  of  a  pound  to  shillings  and  pence. 

30.  Ajdd  g  11).  joz.  Jdwt.  3gr.  together. 

31.  From  9  lb  take  JJ. 

-.  A  colonel,  arranging  his  men  in  a  square  battalion,  found 
that  he  had  ;;i  nun  remaining;  but,  increasing  the  rank  and  file 
by  1  soldier,  he  wanted  20  men  to  make  up  the  square.  Of  how 
many  men  did  his  regiment  consist?  Ans.  656. 

83.  How  shall  I  mark  gloves  that  cost  me  80c.  per  pair  so 
that  I  may  discount  33^  per  cent,  from  the  marked  price  and  yet 
gain  2'>  per  cent,  on  the  cost?  Ans.   $1.50. 

31.  Suppose  that  in  a  shower  the  water  falls  to  the  depth  of 
2  inches,  how  many  gallons  will  fall  upon  a  township  that  is  6 
miles  square,  each  gallon  containing  231  cubic  inches? 

35.  How  many  bricks  8'  long,  4'  wide,  and  2'  thick,  will  be 
required  to  build  a  house  32ft.  long,  24§ft.  wide,  and  20ft.  high, 
the  walls  being  1ft.  4'  thick,  the  house  having  2  doors,  each  4ft. 
wide  and  8ft.  high,  and  21  windows,  each  3ft.  wide  and  Cfu  high, 
no  allowance  being  made  for  the  space  occupied  by  the  mortar? 

36.  What  is  the  square  root  of  the  square  root  of  16  times  81  ? 


MISCELLANEOUS  EXAMPLES.  809 

37.  If  ■  horse  travels  C>\  vaSkt  pet  hour,  how  many  hours  will 
it  take  bim  t<>  travel  m  tar  m  i  rail  car  will  run  in  G  boos 

car  running  82]  miles  pet  hour? 

38.  Light  moves  about  192000  miles  per  second  and 
about  11  12  second;  what  is  the  ratio  of  tin*  v.l-.city  of 
light  to  that  of  nood?  Ans.  887705fff. 

30.  What  is  the  square  root  of  4  times  the  square  of  8  ? 
What  i-  the  cube  of  tin-  square  root  of  j 

41.  What  is  the  cube  root  of  the  square  of  8? 

42.  What  ia  the  square  of  the  culx'  root  of  8? 

43.  Two  ships  sail  from  the  same  port,  one  due  north  and  the 
other  due  west,  one  at  the  rate  of  G  miles  and  the  other  8 

per  hour.     Suppose  the  surface  of  the  ocean  to  be  plane,  how 
far  Apart  are  the  ships  in  10  hours  ? 

44.  An  army  consists  of  59049  men;  how  many  shall  be 
placed  in  rank  and  file  to  form  them  into  a  square? 

1">.  "What  is  the  diameter  of  a  circular  pond  which  shall  con- 
tain 36  times  as  much  area  as  one  20  rods  in  diam 

46.  What  is  the  mean  proportional  between  16  and  6 

47.  What  is  the  third  proportional  to  3  and  30? 

48.  A  ladder  41  feet  long,  will  reach  a  window  40  feet  Ugh 
on  one  side  of  a  street,  and,  without  moving  the  foot,  it  will  reach 
a  window  9  feet  high  on  the  other  side;  how  wide  is  the  street? 

Ans.  49ft. 

49.  What  is  the  difference  in  the  expense  of  fencing  a  circular 
40-acre  lot  and  one  of  the  same  area  in  a  square  form,  the  fence 
costing  50c.  per  rod  ? 

50.  Sold  to  J.  P.  F.  goods  as  follows : 

Jan.  18,  1862,  on  6m.,  75yd.  of  cloth,  at  $4,  $800, 
Mar.  12,  "  "  3m.,  600gal.  of  molasses,  "  33&c,  200. 
June  15,      "        "4m.,     50bbl.  of  flour,         "  $8.  400. 

Also  bought  of  him  : 

Feb.  18,  1862,  on  4m.,  30c  of  wood,  at  $  6,         $180. 

Mav   24,                *    6m.,  I0L  of  hav,  120. 

July     6,      "        "    5m.,  10  o  u  30,  300. 

■      24,      "        "    4m.,     1  ho  ■  100. 

Wh^n  shall  he  pay  me  the  balance  of  the  debt  ? 


310  MISCELLANI.  MI'LES. 

51.  What  is  the  side  of  a  square  equivalent  in  area  to  a  rec- 
tangular field,  which  is  81  rods  long  and  49  rods  wide! 

52.  Sent  an  invoice  of  goods  to  my  agent  in  Liverpool  which 
he  sold  for  $2.3000 ;  what  sura  can  he  invest  for  me,  his  commis- 
sion for  selling  being  2  per  cent,  and  for  investing  1  per  cent.  ? 

53.  A  house  worth  $8000  is  insured  for  I  its  value  ;  what  b 
the  premium  at  g  of  1  per  cent.  ? 

54.  Wli.it   is  the  amount  of  $325,  at  6  per  cent,  compound 
st,  for3yr.  8m.  12d.? 

55.  $1200.  Boston,  Mmj  12,  lft  60. 

:•  value  received  of  A.  B.  I  promix-  t<»  pay  him,  or  his 
order,  one  thousand  two  hundred  dollars,  on  demand,  with 
inter  Chaki.i.s    D\\r. 

Ini.mi:-  .;0,  $300  :    [fee.    1*,  1860,  $10; 

May  6,  1861,  $16.50;  Jure    Jl.    1861,  S100;   Dee.  21,   1861, 
$100 ;  what  was  due  Apr.  12,  1# 

56.  A  bushel  mra-ure  is  18^  inch«s  in  diameter  and  ft  inches 
d« •.•]> ;  what  are  the  dimensions  of  a  nmilar  measure  that  holds 
halt"  a  peek  ?  An-.   9}in.  diameter;  4in.  deep. 

57.  Sold  a  lot  of  goods  for  $100  and  thereby  gained  25  per 
cent.  ;  what  per  emt.  >hould  I  have  gained,  had  I  sold  them  for 
$12< 

58.  A  garden  whose  breadth  is  5  rod*,  and  whose  length  is 
1$  times  its  I»n  a-lth,  has  a  wall  Z\  feet  thick  and  4  feet  high, 
around  it,  outride  of  the  line;  what  was  the  cost  of  this  wall  at 
3£c.  per  cubic  foot  ? 

59.  What  will  be  the  cost  of  digging  a  ditch  around  the  above- 
mentioned  garden,  within  and  adjacent  feo  the  wall  3£  feet  wide 
and  2^  feet  deep,  at  $  of  a  cent  per  cubic  foot? 

60.  What  would  be  the  cost  of  walling  the  above-mentioned 
garden,  the  central  line  of  the  wall  to  be  on  the  bounding  line, 
Ike  wall  to  be  3^  feet  thick  and  3J  feet  high  and  to  cost  G£c.  per 
cubic  foot  ? 

1 1 .  A  hare  has  45  rods  the  start  of  a  hound,  but  the  hound 
runs  12  rods  while  the  hare  runs  9  ;  how  many  rods  will  the  hare 
run  before  the  hound  overtakes  him  ? 

62.  A  hare  has  32  rods  the  start  of  a  hound,  but  the  hare  runs 


only   :  iiile   the   hound   runs  20 ;   how  far  will  the  hound 

run  before  be  overtake!  the  1 

the  inter.  JO  from  Aug.  8,  18G1,  to  July 

A.  K  and  (' engage  to  do  a  piece  of  work]    A    DM    do   it 

•i  2  1,   and  C  in  30.      In  what  time   ean   the  time 

together  do  the  work  ? 

\     .nth-man  left  his  Bon  an  i  f  which  be  .-|»ent  in 

UP  and  {\2  of  the  remainder  in  •)  months   nioiv,  when  bl 
only  $!  Ion  remaining  ;   what  was  tin-  value  of  lL 

66.   The  I'lHiiiiKiiiilcr  of  a   he>ieged   fbftCMI  bai  2lh.  of*  hnad 
pOf  day  tor  each    -oldier   for  46   day-,  hut  wi-hrs  to  prolofl 
60  days  ;   what  DDOSt  he  the  allowance  per  day? 
G7.   A  man    Bold   a    wateh    for    >»)<»,    which    w;u>  j  of  its    cost; 
what  was  lost  hy  the  transaction? 

It' a  btt  of  silver  1ft.  f>in.  long,  tin.  wideband  2in.  thick,  is 
worth  11240,  what  is  the  value  of  a  har  of  gold  1ft.  3in.  long 
8in.  wide,  and  lin.  thick,  the  weight  of  a  euhic  im  h  of  .-ilvrr 
;  to  the  weight  of  a  euhic  inch  of  gold  as  10  to  10,  and  tho 
value  per  ounce  of  silver  being  to  that  of  gold  ■  2  to  3.'3  ? 
69.  dan.  1,  1861,  A,  I>,  and  C  form  a  partnership  for  1 
and  each  furnishes  $20<»  1,  A  furnishes  $1000  more; 

June  1,  B  furnishes  SloOO  and  C  withdraws  $500;  Oct.  1.  A 
withdraws   $500,  and    IS   and    Q  furnish   $1000  each.      II 
gained  $3000,  at  the  close  of  the  year  the  partnership  i- 
solved.     What  is  *-aeh  partner's  share  of  the  gain  ? 

7<>.    How  many  gallons  of  wine  at  6,  10,  l.">,  and  20s.  per  gal. 
may  he  taken  to  form  a  mixture  of  (J5  gallons  worth  12 
gallon  ? 

71.  Find  the  difference  in  time  due  to  a  difference  of  17°  20' 
40"  in  longitude. 

72.  The   difference   in    the   time  of  two  places   is   3h.    18m. 
15eec;   what  i-  the  difference  in  longitude? 

A    merchant    hought    a    numher  of  ha 

the  rate  of  $7  and  sold  them 

out  at  the  for  7\.l  ,  and  gained  $200  by  the  bargains; 

An-.  0. 


miscellan: 

71.  The  trans-Atlantic  telegraph  laid  in  1S">7  from  St.  John's, 
foondland,  to  Valentia,  Ireland,  1640  miles  En  a  straight  line, 
consisted  of  7  copper  wires,  twisted  together,  imbedded  in  prima 
percha,  and  surrounded  by  18  bundles  of  iron  wire  Bach  bun- 
of  iron  wire  consisted  of  7  wires  which  were  twisted  together, 
and  the  bandies  ran  spirally  round  the  cable.  Now,  to  allow  lor 
deviations  ftw  ;hfl  coarse,  ineqnaUttea  of  die  sea-bottom, 

1 1 1  tunes  as  l«>iig  as  would  be  required 
for  a  straight  coarse,  ami  that  it  was  necessary  to  increase  the 

wiVfl    1    Bails    in  every  20   in    con-eqnence  of  twisting  the  wires, 
and  1  milr  in  eve: •_.  uise  of  the  bundles   running  spirally, 

what  length  of  wire  was  required  faff  the  cable? 

0905)  ni: 

7').  By  the  census  of  1860,  the  number  of  inhabitants  of  Ala- 
bam..  J96;  ofArka:  Vtt  |  of  California,  880016] 
of  (  1;  of  Delaware,  112218;  of  Florida, 
140439]  of  G  I  of  Illinois  1711753]  Of  Indi- 
ana. 1350941  .  1  <>71 10;  of  Ken- 
tucky, 11557131  of  Louisiana,  700290;  of  Maine,  528276;  of 
Man  lane                   !  ;    of   Ma  of  Mich 

749111;  of  Minnesota,    172022;  of  Mississippi,  ;  of 

JOOri,  1182317;  of  New  Hampshire,  .120)72;  of  New  Jeff- 

672031  ;    of   New    York,    3880735  ;  of   North    Carolina, 
992667  ;  of  Ohio,   2339599  ;  of  Or-  of  Pennsylva- 

nia, 2906370;  of   Rhode  Island,  174621;  of   South   Carolina, 
703812;  of  Tennessee,   1109847;  of   T  -132;  of  \ 

monr.  B)  of   Virginia,  1596079;  of  Wisconsin,  775873; 

of   the    District  of   Columbia,   75076;  and  of  the   Territories, 
220143  ;  what  was  the  population  of  the  United  States  in  1860  ? 

Ans.  31443790. 


STANDARD  ARITHMETICS. 

EATON'S   COMPLETE   SERIES, 

VVTEI)    TO    TUT      BEST     u  ;»'H:     OF     INSTKHTIOV 

I. 

TIT 
IV. 


THE    PRIMATT  TIC 


THE  IJS 


THE  NEW   TREATISE-  ON   WRITTEN   ARITHMETIC 

;i£,^H  the 

■ 


THOMPSON, 

i>ul  Ushers,  29  CoraMll,  B6ston. 


